Stochastic Integrals: Proceedings of the LMS Durham Symposium, July 7-17, 1980 (Lecture Notes in Mathematics, 851) 3540106901, 9783540106906
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continuation on page 541
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
851
Stochastic Integrals Proceedings of the LMS Durham Symposium, July 7 - 17, 1980
Edited by D. Williams
Springer-Verlag Berlin Heidelberg New York 1981
Editor
David Williams Department of Pure Mathematics, University College of Swansea Singleton Park, Swansea SA2 8PP, Wales, United Kingdom
AMS Subject Classifications (1980): 33-XX, 35-XX, 53-XX, 60-XX, 81-XX ISBN 3-540-10690-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10690-1 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
0
1 •
There are many people and organisations to thank, including:
The London Mathematical Society;
and especially Heini Halberstam, John Williamson,
and Tom Willmore, for 'pre-natal' care on behalf of LMS; The Science Research Council, for generous financial support;
and especially
John Kingman, SRC assessor for this symposium, for his interest and sound advice; The Durham Mathematics Department;
and especially Tom Willmore, Peter Green, and
of course, Ed. Corrigan; Grey College, myoId home, for its usual warm hospitality;
Paul-Andre Meyer, for several valuable suggestions; My wife Sheila, and her father, the late great Edward Harrison, for a lot of work and for unlimited patience; Robert Elliott, my co-organiser, for effective troubleshooting in some moments of minor crisis, and for much helpfulness throughout two years; And Chris Rogers and Margaret Brook, whose very hard work somehow defeated my determined efforts to surpass Haydn, Wiener, and
in achieving a representation
of chaos.
But, above all, thanks are due to all participants:
for a marvellous time;
for fine mathematics; and, no less importantly, for fun and friendship.
2
0
•
The three introductory articles - by Elliott, Rogers, and myself -
are intended to help make some of the later material accessible to a wider audience. At the symposium, there was much interest in the Malliavin calculus. My introductory effort is intended to provide some background material for this topic and for related topics.
David Williams
PARTICIPANTS (WITH ADDRESSES) Convention: CAMBRIDGE = Department of Pure Mathematics and Mathematical Statistics; University of Cambridge; 16 Mill Lane; CAMBRIDGE CB2 1SB; England. HULL = The University of HUll; 22 Newland Park; Cottingham Road; HULL HU6 2DW; England. PARIS VI = Laboratoire de calcul des probabilites; Universite de Paris VI; 4, place Jussieu, Tour 56; 75230, PARIS Cedex 05; France. STRASBOURG = Departement de Mathematique; Universite Louis Pasteur de Strasbourg; 7, rue Rene Descartes; 67084 STRASBOURG, France. SWANSEA = University College of Swansea; Singleton Park; SWANSEA SA2 8PP; Wales. United Kingdom.
L. ACCARDI; Istituto Matematico Federico Enriques; Universita di Milano; Via L. Cicognara; 20129 MILANO, Italy. S. ALBEVERIO; Institut fUr Mathematik; GebKude NA; UniversitKtsstr.150; Postfach 2148; 463 BOCHUM; W. Germany. D.J. ALDOUS; Department of Statistics; University of California; BERKELEY; California 94720, U.S.A .. A.N. AL-HUSSAINI; Department of Mathematics; The University of Alberta; EDMONTON T6G 2G1; Canada. J. AZEMA;
PARIS VI.
A.J. BADDELEY; A. BARBOUR;
CAMBRIDGE.
M.T. BARLOW; T. BARTH;
CAMBRIDGE.
LIVERPOOL - now at CAMBRIDGE.
Department of Pure Mathematics; HULL.
J.A. BATHER; School of Mathematical and Physical Sciences; The University of Sussex; Falmer; BRIGHTON BN19QH; England. P. BAXENDALE; Department of Mathematics; King's College; University of Aberdeen; High Street, ABERDEEN AB9 2UB; Scotland, United Kingdom. D. BELL;
Department of Pure Mathematics; HULL.
K. BICHTELER; Department of Mathematics; University of Texas at Austin; AUSTIN; Texas 78712; U.S.A. N.H. BINGHAM; Department of Mathematics; Westfield College; Kidderpore Avenue; LONDON NW3 7ST; England. J.M. BISMUT; Departement de Mathematique; Universite de Paris-Sud; ORSAY 91405; Paris, France.
v T.C. BROWN; School of Mathematics; University of Bath; Claverton Down; BATH BA2 7AY; England. T.K. CARNE;
CAMBRIDGE.
Mireille CHALEYAT-MAUREL;
PARIS VI.
L. CHEVALIER; Laboratoire de Mathematiques Pures; Institut Fourier, Universite de Grenoble; B.P. 116-38402 Saint Martin GRENOBLE, France. J.M.C. CLARK; Department of Computing and Control; Imperial College; 180 Queen's Gate; LONDON SW7 2BZ; England. R.W.R. DARLING; England.
Mathematics Institute; University of WARWICK; COVENTRY CV4 7AL;
A.M. DAVIE; Department of Mathematics; University of Edinburgh; James Clarke Maxwell Building; The King's Buildings; Mayfield Road; EDINBURGH EH9 3JZ; Scotland, U.K. M.H.A. DAVIS; Department of Computing and Control; Imperial College, 180 Queen's Gate; LONDON SW7 2BZ; England. C. DELLACHERIE; Department de Mathematique; Universite de ROUEN; B.P. No. 67; 76130 MONT-SAINT-AIGNAN; Rouen; France. R.A. DONEY; The Manchester-Sheffield School of Probability and Statistics; Statistical Laboratory; Department of Mathematics; The University, MANCH8STER M13 9PL; England. H. DOSS;
PARIS VI.
E.B. DYNKIN; Department of Mathematics; CORNELL University; White Hall, Ithaca; NEW YORK 14853; U.S.A. D.A. EDWARDS; Mathematical Institute; University of Oxford; 24-29 St. Giles; OXFORD OX1 3LB; England. R.J. ELLIOTT;
Department of Pure Mathematics; HULL.
K.D. ELWORTHY; England. P. EMBRECHTS; Belgium. M. EMERY; H. FOLLMER;
Mathematical Institute; University of WARWICK; COVENTRY CV4 7AL; Departement Wiskunde KUL; Celestijnenlaan 200-B;
B-3030 HEVERLEE,
STRASBOURG. Mathematik; ETH-Zentrum; CH-8092 ZURICH; Switzerland.
M. FUKUSHIMA; College of General Education; Osaka University; 1-1 Machikanayamacho; Toyonaka-shi; OSAKA 560; Japan. D.J.H. GARLING;
CAMBRIDGE.
G.R. GRIMMETT; School of Mathematics; University of Bristol; University Walk; BRISTOL BS8 1TW.
VI B. HAJEK; Coordinated Science Laboratory; College of Engineering; University of Illinois at Urbana-Champaign; URBANA, Illinois 61801; U.S.A. J.M. HAMMERSLEY; Institute of Economics and Statistics; University of Oxford; St. Cross Building; Manor Road; OXFORD OX1 3UL; England. J. HAWKES;
Department of Statistics; SWANSEA.
R. HOLLEY; Department of Mathematics; University of Colorado; Boulder; COLORADO 80309; U.S.A. M. JACOBSEN, Institute of Mathematical Statistics; University of Copenhagen; 5 Universitetsparken, DK-2100, COPENHAGEN Denmark.
¢'
J. JACOD: Laboratoire de Probabilites (C.N.R.S.), Universite de Rennes; Avenue du General Leclerc; Rennes Beaulieu, 35042 RENNES Cedex; France. T. JEULIN;
PARIS VI.
K. JANSSEN;
DUSSELDORF.
KARKYACHARIAN;
Universite de NANCY; France.
D.G. KENDALL;
CAMBRIDGE.
W.S. KENDALL,
Department of Mathematical Statistics, HULL.
H. KESTEN, Department of Mathematics; White Hall; CORNELL University; Ithaca; NEW YORK 14853, U.S.A. P.E. KOPP;
Department of Pure Mathematics; HULL.
P. KOTELENEZ; Forschungsschwerpunkt Dynamische Systeme; Bremen, Bibliothekstrasse, Postfach 330440; 2800 BREMEN 33; West Germany. H. KUNITA; Department of Applied Science; Faculty of Engineering; KYUSHU University; Hakozaki; FUKUOKA 812; Japan. A. KUSSMAUL, Mathematisches Institut der 10; 7400 TUBINGEN 1; West Germany.
TUbingen- Auf der Morgenstelle
E. LENGLART; Departement de Mathematique; Universite de ROUEN; B.P. no.67; 76130 MONT-SAINT-AIGNAN; Rouen, France. J.T. LEWIS; School of Theoretical Physics; Dublin Institute for Advanced Studies; . 10, Burlington Road; DUBLIN 4; Eire. T. LYONS; Mathematical Institute; University of Oxford; 24-29 St. Giles; OXFORD OX1 3LB; England. P. MALLIAVIN; Departement de Mathematique; Universite de Paris VI, 4 place Jussieu, Tour 56; 75230 PARIS Cedex 05; France. P. McGILL; Department of Mathematics; University of Ulster; Northern Ireland; U.K.
COLERAINE;
VII
P.A. MEYER;
STRASBOURG.
S. MOHAMMED; J. NEVEU;
School of Mathematical Sciences, University of KHARTOUM, Sudan.
PARIS VI.
F. PAPANGELOU; The Manchester-Sheffield School of Probability and Statistics; Statistical Laboratory; Department of Mathematics; The University; MANCHESTER M13 9PL; England. J. PELLAUMAIL; France.
I.N.S.A.; 20 Avenue des Buttes de
B.P. l4A; 35031 RENNES;
M. PINSKY; Department of Mathematics; College of Arts and Sciences; NORTHWESTERN University; Evanston; ILLINIOS 60201; U.S.A. G.C. PRICE;
Department of Pure Mathematics; SWANSEA.
P. PROTTER; Department of Mathematics and Statistics; Lafayette; IND 47907; U.S.A.
PURDUE University;
B. RIPLEY; Department of Mathematics; Huxley Building; Imperial College; Huxley Building; Imperial College; 180 Queen's Gate; LONDON SW7 2BZ; England. L.C.G. ROGERS; SWANSEA - now at Department of Statistics; University of WARWICK; COVENTRY CV4 7AL; England. M.J. SHARPE; Department of Mathematics; University of California, SAN DIEGO; P.O.Box 109; LA JOLLA, California 92093; U.S.A. R.F. STREATER; Department of Mathematics; Bedford College; University of London; Regents Park; LONDON NWl 4NS; England. C. STRICKER;
STRASBOURG.
D.W. STROOCK, Department of Mathematics; University of Colorado; Boulder; COLORADO 80309; U.S.A. J.C. TAYLOR; Department of Mathematics; Burnside Hall; 805 Sherbrooke Street West; MONTREAL PQ; Canada H3A 2K6. L.C. THOMAS; Department of Decision Theory; University of Manchester; MANCHESTER Ml3 9PL; England. G. VINCENT-SMITH; Mathematical Institute; University of Oxford; 24-29 St Giles; OXFORD OXI 3LB; England. J.B. WALSH; Department of Mathematics; University of BRITISH COLUMBIA; 2075 Westbrook Hall; VANCOUVER, B.C. V6T IW5; Canada. S. WATANABE; Department of Mathematics; Faculty of Science; Kyoto University; KYOTO; Japan. J. WATKINS;
Free University of BERLIN.
VIII
D. WILLIAMS;
Department of Pure Mathematics; SWANSEA.
T.J. WILLMORE; Department of Mathematics, University of Durham, Science Laboratories; South Road, DURHAM DHI 3LE. E. WONG; Department of Electrical Engineering and Computer Sciences; University of California, BERKELEY, California 94720; U.S.A. M. YOR;
PARIS VI.
M. ZAKAI; Department of Electrical Engineering; TECHNION-Israel Institute of Technology; TECHNION CITY; HAIFA 32000; Israel.
CONTENTS
Introductory articles David Williams: L.C.G. ROGERS:
1
"To begin at the beginning: Stochastic integrals:
Robert J. ELLIOTT:
56
basic theory
Stochastic integration and discontinuous martingales
72
Papers based on main talks and courses Sergio ALBEVERIO and Raphael H@EGH-KROHN: Some Markov processes and Markov 497 fields in quantum theory, group theory, hydrodynamics, and (late entry) C·-algebras Jean-Michel BISMUT: theorem
Martingales, the Malliavin calculus, and Htlrmander's
85
M. FUKUSHIMA: On a representation of local martingale additive functionals of symmetric diffusions
110
Bruce HAJEK and Eugene WONG: stochastic integration
Set-parametered martingales and multiple
119
R. HOLLEY and D. STROOCK: Generalised Ornstein-Uhlenbeck processes as limits of interacting systems
152
Jean JACOD and Jean MEMIN: Weak and strong solutions of stochastic differential equations: Existence and stability
169
Hiroshi KUNITA: On the decomposition of solutions of stochastic differential equations
213
P.A. MEYER:
"
A differential geometric formalism for the Ito calculus
Mark A. PINSKY:
Homogenization and stochastic parallel displacement
Jim PITMAN and Marc YOR: R.F. STREATER:
Bessel processes and infinitely divisible laws
Euclidean quantum mechanics and stochastic integrals
Daniel W. STROOCK:
The Malliavin calculus and its applications
Y. TAKAHASHI and S. WATANABE: The probability functionals (OnsagerMachlup functions) of diffusion processes
256 271
285 371
394 433
Papers based on splinter-group talks Ata AL-HUSSAINI and Robert J. ELLIOTT: two parameter processes L. CHEVALIER: H. FOLLMER: W.S. KENDALL:
"
Ito and Girsanov formulae for
LP-inequalities for two-parameter martingales Dirichlet processes Brownian motion, negative curvature, and harmonic maps
464 470 476 479
P. KOTELENEZ: Local behaviour of Hilbert space valued stochastic integrals, 492 and the continuity of mild solutions of stochastic evolution equations
"TO BEGIN AT THE BEGINNING: by David Williams
Some readers may be helped by this more-or-less self-contained introduction to some important concepts: stochastic integrals;
continuous semimartingales and the associated
the Stroock-Varadhan theorem and its consequences for
martingale representation;
and, as a main theme,
the Girsanov theorem;
the modern theory of the Kolmogorov forward (or Fokker-Planck) equation, involving hypoellipticity and all that, Comments on notation and terminology. to be equal to', differentiable
, ,
-
The symbol
signifies 'is defined
By a smooth function we shall always mean an infinitely function.
We use
to denote the space of smooth
functions of compact support. The summation convention is used throughout the paper, so that, for example, in equation (1.3), it is understood that the first term on the right-hand side is summed over the (repeated) indices summed over
1.
i
and
j,
while the last term is
Note especially:
A brief Appendix at the end of this paper collects some information about Schwartz distributions and hypoelliptic operators,
Part I,
Fifty years of the forward equation,
I think it best to start by trying to motivate things via this account of various approaches to diffusion theory even though it means speaking of certain concepts before recalling their definitions.
KOLMOGOROV (1931),
mn is a path-continuous process and
*
Roughly speaking, a diffusion process 1 ,X 2 ,.,.,X n) X = (X
such that for
X t
h > 0, The 'one up, one down' convention does not work well for transposes:
on 0
2
( 1.1)
for some functions some functions
a
b
ij
Note that for each
i
(1
(1 x
i
n)
n)
i, j in
m,n
on on
called 'drift coefficients', and n
m
the matrix
[Let me mention one technical difficulty:
called 'diffusion coefficients'. a(x)
I have stated (1.1), the integrals
as
determining the expectations could blow up;
is positive semi-definite.
I skip this
so we need to truncate.
now because it is subsumed and superceded via the later use of
martingales.]
Various heuristic arguments (turned into precise proofs and theorems below) suggest that, under suitable conditions, a transition density function
p[X where
p
u+t
Pt(x,y):
E dYlx :s < u; X s
X must be a Markov process possessing
-
u
= x]
satisfies the Kolmogorov backward and forward equations now to be
described.
Let
9
be the operator defined as follows:
(9f)(x) If, for example, the functions in
a
ij
and
i
are smooth, then for
we have
J
h(x)(gf)(x)dx
mn
where
b
J
f(x)(9*h)(x)dx,
nt
is the adjoint operator with
Then the Kolmogorov backward and forward equations take the form:
(B) (F)
f
and
h
3
9
The subscripts on
these operators act; saying that
p
9*
and
x
are meant to indicate the variables in which
y
but it is neater to speak of the forward equation by
P.(x,.)
satisfies
(F) (As usual,
P.(x,.)
is the function
(t,y)
Pt(x,y).)
The early work of Kolmogorov, Feller, and others used partial-differentialequation (POE) theory to establish (under suitable conditions) the existence of a Markov transition density function of
p
satisfying
X, as a process 'proper' carried by some
(B)
(O,d,P),
and (F);
the existence
could then be deduced from
the Kolmogorov-Daniell theorem supplemented by Kolmogorov's criterion for path continuity.
STROOCK-VARADHAN
(1969).
We jump on to the Stroock-Varadhan approach
because it exactly captures the spirit of
(1.1).
The point is that
(1.1)
may be
formulated precisely as follows:
X1.
_
X1.
-
Jt b i (X )ds s
too
defines a local martingale
(2.1) t a i j ( x )dS s Jo The (generalised)
is a local martingale,
formula implies that the conditions
(2.1)
are exactly
equivalent to the following statement:
(SV)
f(X ) - f(X ) - Jt(9f)(X )ds
too
s
defines a martingale
Stroock and Varadhan make (SV) the defining condition for a diffusion process.
One advantage is clear:
if
9
is a second-order elliptic operator
on a manifold, then (SV) makes perfect sense as a condition on a process values in the manifold.
X
f
C •
with
4
Let us be more speci fic about the Stroock·-Yaradhan approach. the manifold in
n
m.
n
Il
We now insist that
Thus, let
of subsets of
w S
Fix
x E m
n•
C
f
x
w
[0,(0)
from
the smallest G-algebra
X (w) s
measurable all maps
s
with
t;
00].
Let
a probability measure starting from
Define
which makes
A = G!X :s
0 T (4.1) J0 5 (x) > < ",* s (x ) ds = C(w
(cp*
is a positive definite form on Tx (Rd). Proofl This result is an extension etMalliavin • Let Us be the vector space in Tx(R d) spanned by -lXi (x)} 10;; io;; m and Vs
{cp'\
be the vector space
(4.2)
u (U t) to;; s
Vs
We now define (4-3)
v:
v:
by
n Vi; t>s
By the zero-one law, we know that depending on
w. Assume that
v+o I
is a.s. a fixed space not Tx (R d). If
T is
the stopping
96
time
*
a. s. T is> 0 • Let f be a non zero element in T (R d ) orthogonal x ,.J to On [0,':eI , we have (4.5)
) -
E(.b(C( w »f(CfJ T(w ,
v li)
dSCfJ"'s-lXj.
s i T ' v J
< C-l(w )cp'" -ly 'PO!.
ds+
J0
• dv +
-lX.;:pO!. -1 X.] J' s
s
[cp"'v-1Xj'
fo
T
ds(
J0
s
0)
Proof: (4412) is a consequence of Theorem 3.1 and of the Remark which follows this Theorem. In fact, as noted in this Remark, it is feasible to take h in (303) to be (4.13) h
= b(C(w
»
C-l(w )cp\-ly
98
To compute the r.h.s. of (3.3), assume for the moment that Xo ••• Xm have compact support. Borel and bounded defined on [ O,T J with values in Rm, consider the stochastic differential equation
= Xo(x) x(O) = x
(4.14)
dx
dt + Xi(x) (dwi +
and the associated dy
equation (1.5)
= ( E So g(Yt)dt has a density oa Rd+P • Now (5.27) E
So
T
g(; t( w,(x,O») dt = E
J0
+CXl
2
1 s t s;T gv (Yt)dt
so that the measure (5.27) has a density. Since
riP
1>( w,{x,O»
"" Cfl trW ,x), the result is proved.
Remark:The detailed proof may be found in
•
108
REFERENCES
BAXENDALE P., Wiener processes on Manifolds of maps, J. of Diff. Geometry, to appear. BISMUT J .M-., Principes de mecanique aleatotre , to appear. BISMUT J.M., Flots stochastiques et formule de Ito-Stratonovitch generalisee, CRAS 290,483-486 (1980).
[4 J
[5 )
BISMUT J. M., A generalized formula of Ito on stochastic flows. to appear. BISMUT J .M ., An introductory approach to duality in optimal Stochastic control, SlAM Review 20 (1978), 62-78.
[6 J
CLAR K J. M. C ., The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970), 1282-1295, 42 (1971), 1778. DELLACHERIE C., MEYER P.A., Probabilites et Potentiels, chap. I-IV, Paris, Hermann -1975, chap. V-VIII, Paris, Hermann 1980.
[8J
ELWORTHY K.O., Stochastic dynamical systems and their flows, Stochastic analysis, A. Friedman and M. Pinsky ed. pp 79-95, New York Acad , Press 1978.
[9J
HAUSSMAl':N li., Functionals of Ito processes as stochastic integrals, SIAM J. Control and Opt. 16 (1978), 252-269.
[10 J MALLIA VIN P. Stochastic CalCUl
1l 3 of variations and hypoelliptic operators. Proceedings of the International Conference on Stochastic differential equations of Kyoto 1976), pp 195-263, Tokyo: Kinokuniya and New-York: Wiley 1978.
[11
J
[ 12 ]
MALLIA \;1N P., Ck-hypoellipticity with degeneracy, Stochastic Analysis. A. Friedman and M. Pinsky ed., pp 199-214, New-York and London Acad . Press 1978. S TRO()C K D., The Malliavin calculus and its application to second order parabolic differential equations, Prepr-int 1980.
[ 13 J S TROOCK D. W. and VARADHAN S. R . S ., Multidimensional diffusion processes, Grundlehren del' Mathernati schen Wissenschaften, Berlin-Heidelberg-New York, Springer 1979. [ 14 ]
[15
J
JACOD J. and YOR :vi.. Etude des solutions extremales et representation integr-ale des solutions pour certains proolemes de martingales. Zeitschrift Wahrscheinlich keitstheorie verw. Gebiete 38 (1977) 83-125. HOR1vlANDER L. , Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171.
109
HORMANDER L. and MELIN A., Free systems of vector fields, Arkiv for Math. 16 (1978), 83-88.
[17J
ROTHSCHILD L.P. and STEIN E.M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. ABRAHAM R. and MARSDEN J., Foundations of mechanics, Reading: Benjamin 1978.
[19 J HAUSSMANN U., On the integral representation of functionals of Ito processes, Stochastics 3 (1979), 17-27. ICHIHARA K. and KUNITA H., A classi fication of second order degenerate elliptic operators and its probabilistic characterization, Zeitschrift Wahrscheinlich keitstheorie verw , Gebiete 30 (1974), 235-254. DAVIS M. H .A., Functionals of diffusion processes as stochastic integrals, Math. Proc , Camb. Phil. Soc. 87 (1980), 157-166.
22
BISMUT J .M .. a Martingales, the Malliavin calcuJ.us and hypoellipticity under general
conditions. To appear.
ON A REPRESENTATION OF LOCAL MARTINGALE ADDITIVE FUNCTIONALS OF SYMMETRIC DIFFUSIONS M. Fukushima ;( College of General Education Osaka University Toyonaka, Osaka, Japan
§ 1 Introduction
In studying the absolute continuity of diffusions with respect to Brownian motion, a very important role is played by the following representation of the positive continuous local martingale multiplicative functional L of the t
Brownian motion (Wentzell [7]): d L
t = exp {
t
L oJ L(X)dB s s
i=l
-
1
2"
d
t
I 0J r.rx s )
i=l
2
ds},
where f. are measurable functions with P
t
x
oo We say that an AF (n )
sequence M
E
.
o.n
M is locally in ..
llt,
if there exist a nest {K and a n}
n=1,2, ... , such that
( 2 . 3) PX ( Mt =M(n) t ' Vt
(3.8)
l(f,A) =
Proof:
1 +
1
L
m! [f (·)l(f,S.)
m=l
0
m WJA
For multiple Wiener integrals (C = {all closed sets )}(3.8)
reduces to
(3.9)
l(f,A} = 1 +
I
m=l m.
which is well known [5J.
For the general case, we use (3.6) in (3.9)
and write L(f,A)
=1
ex>
+
L
W(f,A) m
m
L
m=l m. k=l
135
=1 + =1+
1
L kT
co
k=l' co
1
"k
[f
L
co
1
W.(f.S.) j=OJ· J
"k
k
0
W]A
k
L --k' [f L(f.S.)
0
k=l .
W]A
The expansion formula (3.8) for exponentials of the form (3.7) can be extended with the Wiener integral f by a stochastic integral f Proposition 3.2.
0
W.
0
Win the exponent being replaced
The result can be stated as follows:
Equation (3.8) remains valid for f E
such
that f is bounded. Proof:
Define f to be a discrete simple function if f is a simple
function
such that P(aiEJ) = 1 for some finite set J. written as f(t.w) g(t.c) = Then g(·.c)
(3.10)
E
k
I
i=l
Such a function may be
= g(t.a(w)) where a = (a 1 •..•• a k ) and c.I A (t) for c E JK. 1 i
L2(T) for each c
L(g(' .cLA)
1 +
E
Jk so by Corollary 3 of Theorem 3.1.
21 m! [g (·.c)L(g(·.cLs.) 00
1
0
m W ]A'
This equality holds in L2(n.F.p) for each c E JK and hence it continues to hold in L2(n.F.p) if c is replaced by the random vector a(w). By proposition C in appendix C. replacing c by a(w) in the stochastic integrals is equivalent to replacing c by new) in each of the integrands
136
and then forming the stochastic integrals.
(To apply propositon C to
term on the right of (3.10), let Bi = SA.)' 1 equation (3.8) if f is a discrete simple function. the
This verifies
Conclude that E[L(f,A)] = 1 if f is discrete and simple. if p
1 and If(',')1
l(f,A)P
= L(pf,A)
Moreover,
r for some constant r , then exp(t(p2_ p
)! f(t)2 dt)
l(pf,A) so that
(3.11) Now choose any f E La2(Txn) with If(w,t)1 -< r. Then there is a sequence of discrete simple functions f. f in L2(Txn) such that J a Ifj{w,t)! r for each j. Hence (fjoW)A (foW)A a.s. in L2(n) so that taking a subsequence if necessary, we can assume that (fjoW)A with probability one.
Thus L(fj,A)
(foW)A
L(f,A) with probability one.
the estimate (3.11), the collection of random variables {L(f.,A)P:p J
is uniformly integrable for each p l P(n) for each p > 1.
Moreover, T.
>
-
1 so that L(f.,A) J
By > -
l}
L(f,A) in
in LP(T An) for each P > 1 a since these functions are uniformly bounded. Now (3.8) is true for f J
T
replaced by f j, and it is then easily verified for f by taking the limit in L2(n) term by term as j +00. n 4.
A likelihood Ratio Formula let {Zt,t E T} be a bounded process defined on (n,F,p) and let
{W(A) , ACT} be a Wiener process defined on the same space.
= o({W(B), B C A}, {Zt' tEA}). We assume AnA' = ¢ -WeA') in F(A)-independent. For any F(A)
let
that collection C the support
137
St contains t.
Hence, Zt is F(St) measurable.
For any C satisfying
c l - c 3' the stochastic integral Z 0 Wis well-defined. Now, let P' be a measure on (n,F) defined by: dP' = exp{Z --dP
(4.1)
0
W- -21 Z2
and set
For any C satisfying cl - c 3 ' proposition 3.2 yields (4.3)
It follows that L(Z,A) = E(
(4.4)
F(A))
and pI is a probability measure. Next, let FW(A)
=
a({W(B),BCA}), and define the likelihood ratio
by (4.5)
A(A) =
I FW(A))
We shall use (4.3) to derive an expression for A(A). Proposition 4.1.
(4.6)
Z (t) m
Let tErm and define
= E'(Z(t l)Z(t2)··Z(t )IFW(St t m
Then the likelihood ratio is given by
t))
1 2'" m
138
A(A)
(4.7)
Proof:
=1 +
I
m=l m.
[Zm(')A(S . )
0
wm]
We begin by writing
and using (4.3).
Observe that with P-measure 1,
= E[Z(tl)Z(t2)···Z(tm)L(Z,Stlt2··tm)IFw(A)] =
E[Z(t l)Z(t 2)···Z(t )l(Z,St t t )!FW(St t t)] m 1 2'· m 1 2'· m
=
A(St t )E'[Z(tl)···Z(t )!FW(St t)] l"m m l"'m
and (4.7) follows.
n
Two special cases are of particular interest. a fixed unit vector (i.e., DaD {t ERn: (t ,«)
>
-
a}.
First, let a
E
lR n be
= 1) and let Ha denote the half space
Then, the collection C = {H () T} is a one-
parameter family of sets such that
rm
a
is vacuous for m > 1.
That is,
two or more points are always C-dependent. In this case the likelihood ratio formula reduces to A(A) = 1 + [Zl(·)A(S.)
0
W]A
AE C
139
and an application of (3.8) yields
(4.8) where
In this case we see that the likelihood ratio is expressible as an exponential of the conditional mean. The second case of special interest results from taking C = {all closed sets in T}.
For this case
St t t = {t l,t 2,···,t } 1 2··· m m Hence, with P-measure 1
and
-Zm(t) = E' Z(tl) ••• Z(tm)]
Furthermore, if we assume that Z and Ware independent processes under P then Z is identically distributed under P'. can write
{4.9}
A(A)
=1 +
I
m=l m:
where pm is the -mth moment (4.l0)
(p
0
m
\fI)A
Hence, for that case we
140
Equation (4.9) provides a martingale representation of the likelihood ratio for the "additive white Gaussian noise" model under very general conditions.
In the one-dimensional case, it was recently obtained in
[7]. Equation (4.7) is an integral equation in that A occurs on both sides.
In special cases [2,6,9J the equation can be converted to yield
an exponential formula for A in terms of conditional moments.
141
References 1.
Cameron, R. H., Martin, W. T.:
The orthogonal development of non-
linear functionals in a series of FourierHermite functions.
Ann.
of Math. 48, 385392 (1947). 2.
Duncan, T. E.: noise.
3.
Likelihood functions for stochastic signals in white
Inform. Contr.
Hajek, B. E.:
lE.., 303310 (1970).
Stochastic Integration, Markov Property and Measure
Transformation of Random Fields. 4.
Ito, K.: Stochastic integrals.
Ph.D. dissertation, Berkeley, 1979. Proc. Imp. Acad. Tokyo 20, 519524
(1944). 5.
Ito, K.: Multiple Wiener Integral.
J. Math. Soc.
157169
(l951) • 6.
Kailath, T.:
A general likelihoodratio formula for random signals
in Gaussian noise.
IEEE Trans. Inform. Th.
350361 (1969).
7.
Mitter, S. K., Ocone, D.: Multiple integral expansion for nonlinear filtering. Proc. 18t h IEEE Conference on Decision and Control, 1979.
8.
Wong, E., Zakai, M.:
Martingales and Stochastic integrals for pro-
cesses with a multidimensional parameter. Z. Wahrscheinlichkeits theorie 29, 109122 (1974). 9.
Wong, E., Zakai, M.: Likelihood ratios and transormation of probability associated with twoparameter Wiener processes.
Z. Wahrschein
1ichkeitstheorie 40, 283309 (1977). 10. Vor, M.: Representation des martingales de carre integrable relative aux processus de Wiener et de Poisson scheinlichkeitstheorie
121129 (1976).
a n parametres.
Z. Wahr-
142
Appendix A:
Proof that Simple Functions are Dense
The purpose of this appendix is to prove the following proposition: Proposition A.
Conditions c 2 and c3 imply that the space of simple functions is dense in L2(rmxn) for each m > 1. a
-
Proof:
We begin by introducing some additional notation. t = (tl ••.•• t m) E Tm• define the -support of t by
For
>
0 and
where B( .t i) denotes a ball with radius and center tis and define Define LZ(rmxn) the same way as but with St(-) = U £>0
condition hZ replaced by the stronger condition: for each Am Am t E T • ¢t is Finally. let CE(T xn) be the subspace of L2(rmxn) consisting of ¢ E LZ(rmxn) such that ¢(·.w) is continuous on E
rm with
probability one.
Proposition A is a consequence of the following sequence of lemmas.
lemma A.l. Proof:
U l2(rmxn) is dense in
£>0
Let f E
under conditions c2 and c3,
E
be bounded by a constant r
>
O. For any E > O.
there is a Borel measurable mapping u(,.£) of the open set rm into a finite subset of rm such that
-
0 £ the bounded functions in are dense in the lemma is
(T
established. Lemma A.2.
tI
144
Proof:
Let f E
be bounded by some constant r > O.
such that V
VE
0, V(x) : 0 if Ixl
1, and
Choose
f V(x)dx: 1.
mn
For a > 0, define Va E by Vo(x) : and define a function f O on Trn by the convolution: fO(-,w): VO*f(',w) for each fixed w.
rn
C
r"
C
Here the function f(· ,w), which is a priori defined on (IRn)m,;;; IR mn, is extended to a function on all of IR mn by
the convention
: 0 if
Trn.
Note that fO is bounded by rand
sample continuous, and since V(x) : 0 for Ixl
0, fO E C
2E_ O(Trnxn).
Observe that If-foU 2: E[
f If(s) - fO(s)1 2 ds]
Tn
1. Proof:
Consider the following two conditions on C:
(bl)
There is a countable subcollection of 1m which covers Tm a.e.
(b2)
There is a countable subcollection which covers
of disjoint sets in 1m
rID a.e.
By a sequence of lemmas it is shown below that conditions c2 and c3
condition bl
condition b2
the conclusion of Proposition B.
Lemma B.1. m U
U
1=1 TIEP(m) Proof:
2
Let
= (Pl.···Pm)
E E ,
E (Sx)
-
m.
= (ql, •.• ,qm) E T
}
=I
(*)
Choose a permutation
= rr(ql.···.qm) so that for some 1 with 1
1
m,
147
where "P." denotes that p. is to be omitted. 1
1
That is. the permutation
is choosen so that Pl •...• Pi is a minimal set from ql •...•qm with the same support as Pl •...• Pm. ql····qmESn=s
Now Pn+l' ...• P E S m
N
Pl •... •Pi
since
p
Pl •... • To show that R is contained in the left side of (*). it remains to .:L
show that Pl •...• Pi are C-independent.
Now. if Pl •...• Pi were not
then p.E S P for some i. 1 Pl.···.Pi.···. i {AEC: Pl' ...• P E A}
Then
{AEC: Pl •...• Pi' ...• PiEAl.
Intersecting all the sets contained in this collection of sets yields that
s
Pl'·· .• Pi
=
s
Pl •... 'P i •...• Pi
which contradicts our choice of Pl •...• Pi.
Thus Pl •...• Pi are
C-independent so that R. and hence R. is contained in the left side of (*) •
Lemma B.2. Conditions c2 and c 3 imply Condition bl. Proof:
Let
m denote the -subsets of T of the form Alx ... xAm such that.
for some IT E P(m) and some t > O. a)
An ••..•An are C-independent. closed rectangles whose vertices t
have rational coordinates in T C lR n. and
0 is a countable subset of rand Then 1m m
148
m
"9.,
-
m -
U
m 9.,
°{( X,y) : XE T , YE (S ) - }
= U U II R.= 1 IIEP(m)
U
R.=l IIEP(m)
(B.1)
-
IIoS
m,9.,
where
The first term on the right hand side of (B.l) is equal to Tm by lemma B.l. pm(Sm,R.)
Thus, to complete the proof it must be shown that
= 0 for all m
1
and 1 < ! < m.
By Condition c2,
is a closed subset of 19.,
Tm-9., which increases as
decreases to zero. Since S • = FO - U F, it follows that S • is a Borel subset of Tm. m,.. c m,.. By Condition c3, the section
of Sm,! at
has lebesgue measure zero for a.e. y E
Fubini's theorem, jJm(S lemma B.3.
X
m,...) = 0 for
- m.
1 < R.
1 cover Tm a.e .. We claim that for each i > 1 there is a finite collection of disjoint ni sets Di" ... ,D'n in I such that D. = U D... Condition b is then 1 i 1 j=l 1J m satisfied with = {D .. :i > 1,1 < j < n.}. It remains to prove the I
1J
-
-
-
1
claim. By induction, it suffices to establish the cliam for i = 2. .
Fl = Alx ... xA for some Borel sets Al, ...•A CT. Thus, F' m m j=l where Kl •... ,Kr are disjoint and each Kj is the product of m Borel subsets of T.
In fact,
Now
r = U K,
J
is the union of all sets of the form
B1x..• xBm such that B.1 = A.1 or B. = for each i and such that B. = '1 k 11 for at least one i, and these sets are disjoint.
So D2 = .U K. n F2. J=l J The sets Kj n F2. The sets Kj n F2 are disjoint sets in 1m as a required so the claim is established. Lel11l1a 8.4.
Condition b2 implies that the linear span of {lA:AE 1m} is dense in L2(Tm). Flx... xF m where each Fi for any A E 1m and by Condition b2,
Proof:
Let F
IF =
L
d AE1m
lAnF
E
Rn(T). Then A n F E 1m
m. a.e. in T
Since the linear span of functions of the form IF is dense in L2(Tm), the lel11l1a is established.
150
Appendix C Proposition C.
Assume Conditions cl - c3. Let Bl, ... ,Bk be closed subsets of T and suppose that ai(w) is an F(B i) measurable random variable with values in a finite set J for 1 < i < k. Suppose for each C E Jk that h(""c) E h(t,·,c) whenever c i and
and that
= h(t,·,c') a.s.
= ci for all i such that Bi
St.
h(·,·,a(·))ow'" = h(·,·,c)oWml c-a _ () .
Then h(·,·,a(·)) E La(T xn)
a.s.
For each e E {O,l}k, define
Proof:
By condition c2' the set {t:BCS t} is open for each i so that is Borel for each e. Since U = Tm it suffices to prove the lemma when
e
h(t,·,c) for all t.c. and e. = 0 for 1
first
= h(t •.
-
where h(t.w,c) Thus,
Te
Now, for definiteness, suppose that e.1
(kjex)(x - T] (k )
OkEZd
s
- F(T](O')(q»)] ;(dx)dS s
is a
martingale.
is just JotF'
(cp)
The second term on the right side of (cp)ds , the third term converges in
to
(here
term converges to zero. as
m = Jx 2
+
F(Tlt(cp» is a martingale. TJO(cp)
2;(dX)
cp E J(R
r F' (Tl s (cp»T] s (cp)ds t
'0
Q(ex),s.
and
1}
0'
F E
t
Also by the central limit theorem the distribution of m 2\\cpl\;
This determines the weak limit,
In fact, for each
2 (QiJ.)
J0 F" (T] s (cp) )ds
\\q>\\22 m2
is normal with mean 0 and variance
of the
d)
L
), and the fourth
Thus any weak limi t point of {Q (ex) ;
must be such that for all
0'
(1.1)
d
cp
Uh1enbeck process under
) ,
TJt(q»
under any weak limit Q(m), uniquely (see [6]).
is a one dimensional Ornstein -
Q(a)
We still must prove that
{Q(O');O'
1}
is relatively compact.
Before stating the relevant theorem we int,oduce some notation. n
=
(n 1, ••• ,n
d)
and d
(x 1'· •• ,x d) E R where
a = (a
, then
For
1,
••• ,a
d)
are multi-indices and
n "i nd x = Xl , ••• ,x d
q? E
set
and
\\cpl\a n ,
a D =
0 8
1al
x
If
= ad
oX 11 , ••• ,oxd
sUPd' xER
156
(1.2)
Theorem: d
D([O,
(1.13)
>
t ) ...
J
t )
J
Thus the third term on the right side of
(1.7)
converges in
to (1.15 )
-m211Cll1l22
o
s
Thus the weak limit
P
on
d
C([O,=),..P'(R»
> s)ds •
j
(=)
,of the
p(ct),s
such that for all
distribution with mean zero and variance
is the unique measure d
Cll E ,.P(R)
1l 0 ( cp )
has a normal
and for all
F E
•
161
is a
P
(a> )
- martingale.
(T]t(ct'l), ••• ,Tlt(ct'n»
Note that for
is just an
n-dimensional time-inhomogeneous
Brownian motion with covariance
where
p(t)
is non-increasing. In one and two dimensions the diffusion coefficient converges to zero as the time gets large and in three or more dimensions it has a strictly positive limit. We want to emphasize again that even though the limit in each of these examples was Markovian, this is not the usual situation. complicated interactions the limit is hardly ever Markovian.
For more For example
if the interacting system is the stochastic Ising model and we hold the potential fixed except for the external field, then as we vary the external field there is at most one value of the external field for which the limiting process is Markovian.
The Markovian limits obtained above
resulted from the linear nature of the interaction.
This linearity is
one of the clues that the rescaling used in the next section may result in the more interesting limit obtained there.
Another clue th£ the
rescaling in the next section may work is contained in the second moment computation.
From
(1.11)
we see that, if
p
(1. 16)
lim E t ...
d
3 ,
constant d-2
(k)T] t (j) 1
\k _
o:>
as
j!
\k - j \
Every example that we have for which the rescaling in the next section yields an interesting limit satisfies
(1.16).
162
2.
Rescaling Space and Time. In this section we again consider the voter model but use a diffe-
rent rescaling.
Thus
P
will be the same as it was in section
1.
Since we are going to speed up the time we can discover how to renormalize by considering the asymptotic behavior of (2.1) as
0'
lim E t .... cc
')' cp(k/O')ll
k
goes to infinity.
behavior of (2.1)
P
(2.1)
From
is asymptotically
indicates that in
we see that the asymptotic
1
If
whereas i f
f d fdw(x)cp(y) R
3
(1.11)
j
depends on the dimension.
is asymptotically constant
in
t(k»2
2
and
1
Ix
R
-
yl
d
1 or 2
then
d:2: 3
then
(2.1)
dimensions we should divide by d+2
or more dimensions we should divide by
This
d-2 dXdyad+2
If
a
d
and
d = 1 or 2
this yields the uninteresting limit which is identically zero, thus we concentrate on the case d+2 (0')
TJ t
_
(cp) - 0'
-Z\,
I
O')ll
to be determined later. on
d:2: 3
e(O')t(k)
Let
under
For , where
pea) P
i-L
and 13(0')
let
is a function of
be the distribution of and
We use the same notation,
pea) , here that we used in section
The meanings that they had in section
1
even though they are different. 1
will not be used in this
section. Again it is not difficult to show that
{p(a):O':2: I}
is rela-
tively compact by checking the hypotheses of Theorem (1.2), so we will concentrate on identifying the limit. Then
Fix
cp E
and
F E
•
163
d+2
(2 .2)
(cp»
-
So
p(k, j)[ F('ll - T)
S
(a) (qJ)
oo» -
cp(kja)(11 s (j)
F(T)(ja)( )(cp»]dS
Ba
S
-
=
2 +:
d+2
-
- S;F"
(11
s(a)s(j)
- 'D,8(a)s(k»2cp2(kja)a-(d+2)8(a)dS
_.£ - 3 _ 0(0' 2
8(a)t) d+2 -
=
- CP(;))'D,8(a)sU)S(CL)a
r
t 1 (a) " - 2'0 F (11 s
\' " ' .
.
-11
B(a)s(k»
2 2 k cp
2 ds
-d -2 0' e(::;;)ds
- 2. _ 3 - 0(0' 2
8 (O')t)
= F('D,t(O') (cp»
-
t
foF
I
2 (11 (a) (cp»1] (0) (0"2 Llcp)0'-28 (a)ds s s
t
p p(O,j)E \.I[ (11 (O')sU) S
-1,fo F"
'i1
s (a ) s (0»2]0-""2 8 (a)ds
1 st" (0') \' \ ' . . 2 2 1./ -d - 2 0 F (11 s (cp) )[ P (k , J ) ( \ (0' ) s (J) - 1] 8 (a) s (k» cp ('Y 0')1)' P
-
-
is a
8 (0')
0
P =
t1
(a
-2
S (o j t )
- martingale.
i.
2 p (k, j)E \.I[ (11 (I)')s (j) - 11 (O')s (k) )2]cp (kja)O'-d ] 0'-2 8 (I)')ds 8 S
At this point it is clear that we should take
Now consider the third term on tie right side of
(2.2)
164
P
(2.3)
2 (j) -T) 2 (0»
j
Q'S
.... where
p
as
2
] = j
Q'S
Q' .... 00
- 2m 2P
(0
.)
,j
2
(T";Q' s )
,
is the probability that a random walk with transition function
+ p(j,k)] Thus for large
Q'
starting from the origin never returns to the origin. the third term is essentially
Finally we come to the fourth term on the right side of
(2.2).
In all of the examples which we have of this rescaling the computations all have the same ingredients and this term is always the most difficult to handle.
We want to show that it converges to zero in
In this particular case it is not too difficult to show that in fact it converges in
because we have tractable expressions for
the second and fourth moments of On the other hand, even with
T)s(k)
(1.11)
given in
and
(1.11)
and
(1.12).
(1.12) , the computation
is tedious and we will leave it to the reader to check that the fourth term does indeed tend to zero in Note also that
as
E[ g(X\(",.w') =gt(w.X(w,v l one has g(X) E L(Z;n..E:.F) and x: = K + g.
is the "dual predictable projection" of the jump measure of A c JRP
that is for all Borel subsets
v([O,tlxA)
Y
C {!lI Ys l > 1. } + y t
rely discontinuous) local martingale. Next, At last,
Y = (yj) , J";P
= 0,
position:
B
CO=O,
s,,;;t;
and they are characterized as follows: first,
where
?/ 2 •
a positive predictable random measure on JR xlRP .. ith
\1(w;l\>«03> = y(w;(OJ)
(e
i
-1. - iI (I z \
'1.) »)) (ds , dz ) •
Then (c r , [9, proof of 0.51)], or [7J): (2.3) LEMMA: An adapted, right-continuous process
Y
with left-hand limits
is a semimartingale with local characteristics (B,C,v) if and only if, for all u eJRP, the process e i < u] y> _ pu is a local martingale. Now we go back to equation (1.1). In the following we assume (1.2), (1,3), (1.4), and
(B,C,v)
denote the local characteristics of
The following is an increasing predictable process:
Z over
174
We have a factorization
l b
c
= (oj) j""m'
c = (c
Y
(dt,dz)
an ::IRm-valued predictable process,
j k)
k ' an mxm nonnegative symmetric matrix-valued predictaJ, "m ble process, 0
Nt(w,dz),
a predictable transition measure on ::IR
From (2.4) it is easy to see that one may choose
m•
b,c,N
such that
(2.6) Now, define the following collection of predictable processes on - - , with ue:::IRm+d ,and with this additional piece of notation: if -zE::IR m and XE::IR d, we denote by y = (z,x) the vector of ::IRm+d whose components are c
(putting
= zj
if
yj
jk t
L
gj-m,l c lk t t 01 k-m,l gt '[ j-m,l c lq k-m,q l,q""mgt t gt
cg,jk t
(2.7)
yj
=x j-m
otherwise.
if
j,k ;,;:m
if
k ",m
.2!!.
and
l' e
If
with
V(w,x) = U(w)W(x),
we will denote by M (n) =mc
onto
/t of both 11.n= {Pta.: PE,./{J and
ll!!
(3.4) THEOREM:
are continuous.
and PI its marginals on I2. P M-':> Pin. (re sp , l'
M Ul) =m
(3.,3) THEOREM: A subset if
bounded measurable
p!-'L
Of course. the mappings:
are continuous from
i.!!;
U
W bounded uniformly continuous on
(Q)
and on
reV)
is also the coarsest one for which
(resp.
M =c
is relatively compact i f and only J1*,=
are relatively compact
respectively.
(pll)
be a sequence converging to
l'
in
Let
Fw = {XfX: (w,X)E F} .!§. '(s-closed in -n lim sup(n) P (F) = '1. Let V be a bounded measurable
be such that each section .;. O.
be such that
sup(n) sUPs
xnEl':k'
1.s ,
be a sequence of points in
t? 0
s there exists a sequence n Llx (t n ) ------? Ax(t). We have
Since
t !x(s)!) for all
Then:
induce the same topology on l"k' Proof. Let
s
n,gn,U t
/0(t[ vsn,gn"u
,i< I(Zn X K n » ] dAn exp u s-' s- - ss •
We turn now to a first set of conditions. (3.10) Condition on
pn: n
0.11) Condition on
K
(3.12) Condition on
Zn:
:
(pn)
pCO in
converges to
M (12) •
=m
= 0 , all
lim(n)pn[suPs.,;t lim(n)pn
bounded I-stopping time
T
on
n.
I:;. fJ = 0
for all
(3.13) Tightness and linear growth condition: for each a predictable process
"'n
A on and that:
JIll) 0
such that
nEiN
0
and all
there exists
and a predictable increasing process ,vn (t n 2 n At - Jo dAs is increasing
V(os»
(i) (11)
f;>
£.>0.
\x(s)I),
lim(D)pDUs('An,A:co»f)
=0
all
for all
the Skorokhod distance on the space
nelN • .>0,
where
Ss
denotes
D«(O,co),JR+);
(iil) there exists a measurable process
0
on
such that
a s o , wEll the set (t: Vt(£v) such that for all t 0 we have
> a}
is discrete, and
tor all
(3.14) Convergence condition on solution-measures: this is a condition on each pIt being a sOlution-measure of (1.1,n). n co pn£) = 0 for all m+d (n) T _ T £: ';> 0 , u eJR ,and all bounded I-stopping time T on
a sequence We have:
(pn)
0.15) Continuity of
.,.T I
lim
gCO: for all
n..
WEn,
es o ,
is continuous
185
on
endowed with the uniform topology (or equivalently with 00 since gt(G.I,x) depends upon x only through the values x(s), s tJ :IN is tight. Henc e i f "Z /' 0 there d ",mc nE -n exists a bs-compact subset H of such that P (H) '" p(X E H) 't for all Since
nEJN.
Xn
most all
There exists
a'70
such that
xEH
is a solution of (3.28), we also have "'"' E S2-
and all
n EJN.
If
£. '7 0,
n
sUPs,,;t
Xn(w)£*(w)
a. for P-al-
we obtain
,u
X(.)EHnac(.»
"'n vg(X ),u)eXPi £. , s s- sss and Xn(.)EHn*(.)} 3O«'l+lu!2)('l+a) (t('1+0 )Ign(X n) _g (Xn»)dA ;of 1 )0 s s s s J where for the last inequality we have used (2.13), (3.25,i) and the fact Ix(s) 15 a
that
if
XE H.
last term in 0.38) tends to
0
Now, lemma 0.30) implies that the
when
n l oo ]
since "170
is arbitrary,
we obtain (3.14). It remains to prove that if
P
then and the theorem will
is a very good extension of fo110w from (3.16). We may assume that
(pu)
Let
,f,P) ,
M lole a bounded martingale on
=n (p )nEJN'
is a limit point of
(il.,
n ...,
P
tends to s "'"t ,
and
in
UE B
that is F -measurable. Then =:s E[U(M t - Me)] because each surable
U eB
limen) En[U(M t - Ms)]
=:
me
M
=mc
(ft).
(li)
0,
is a very good extension. The set of all F and F=s' =sE(Mt-M IF ) =:0. The right-continuity of M s =sE(M - M IF ) =0, and the result is proved •• t s =s (.IT.)
generates a a--field that is in between
mc so we have proTed that implies that
3-d.
PROOF OF THE MAIN THEOREM (3.16). Let us begin with a "Gronwall lemma", whose proof is reproduced here
for the sake of completeness (see [12J).
191
(3.39) LEMMA: There exists a mapping property:
11
F
(resp.
k:
_lR+
with the following
is a right-continuous nonnegative predicta-
ble (resp. adapted) increasing process on some filtered probability space
11
(Jl.,!,,!:,P),
.!!
FO ",0,
FT
Each stepping time
is predictable, and hence is announced by a sequence stopping times: we have
U(n,j)a) ,
qeJN,
=
Rn(a,a',a",q)
c>(a)
= inf(t:
a'> 0 ,
put
jXtl>a).
we put rn(a)A en(a')f)V
s
secondly the existence of
for at least one
s Condition 0.10) implies the existence of "7
with
Condition 0.13,iii) implies: firstly the existence
such that p(J)(sup
NV
s-.;;N) " ' a for at least one Pn (..1Z s There exists
a' >'1
0.11) there exists
such that
such that: n
s..
I
'i.
4.
(3.51) From 0.47) there exists
ssN)";;
N \K(J) l>a t -'1).,;; £/16. From s n (Xl IK s - Ks -:I.) .:; such that: n
p(J)(sup
£/8. From 0.10) there exists pJL(sUPS:!ON \K: I>a' f/8. Then
qEJN,
such that N)s £/16 3EJN f/8. From 0.10) there exists
and that: such that:
f -:;.0,
=====j>
of
pn(sup
[O,Rn(a,a',a",q»,
s
n
Then
f 4'
195
a),
and consider the Tn,s defined in 0.47). q CD CD -CD We have: lima too cr(a) = (]), and: lim(q) 1Tq =(]) P -a.s., thus P -:.s. Therefore it suffices to prove that, for each fixed a, q and if T = T]lAC) /Irr a ,then q
«])
n
is a
-n
;:::(J)
P
-local martingale, while knowing
P -local martingale for each nEJN. Actually, n one may replace "local martingale" by "martingale", since M is bounded o] by 1.+40 - exp t
E -n
ill c u jYt/lTill»J
ill gill U
CD
E [U(expi 00 n
'En[U(pill,g ,u _ ;nn,g 'U)J. tI\ Too 't" t II Tn n By
T f
T:),
we have
)
2pq luI'
because of (3.11) and (3.20,i).
be a t's-compact subset of H Ign(w,x) _gCD(w,x)l, s
to the measure
s
=
and let
this "ess sup"
boeing taken with respect
:=
Then
for all. xeH, pCD_ a• s• in lAo (use (3.60». By (3.15) and Lebesgue convergence Theorem. we have
-n
P
n -n
CD
-a.s., and (3.55) implies that
p (FH:,;>£)---O.
But
0.
Le t
An = Hn.F.
F be
Set
J
n[n (,n Ht b t Hn [cn,jk + (Nn(dZ)fj(z)fk(z)]
I'n
f
n e IN.
for all
j't
t
n
Nt (dz ) •
t An , I'n An An "'n -I'n B =b .F, C =c .F, -y (h)=N (h),F.
/'
If
uEJR
m
we set
0.63)
/'J'
n U n '1. wt' = i -
hence i f
Cf\i(z) = e
(3 . 64)
Hn
t
i i
n,
Wt
",j
n jk/'k f.n (i A\ Ct' U +jNt(dZ) e -'1-iI{IZI,;;"1.})
A
-'1- Lc u /f(z»
'1.
/'
+2"ku jf(z»)
2
,
we have
A /'b n '1" "'j An, jk "k "'N n (l1J ) - 2" Lj,k.;m U c t U + t 111
•
0.58) implies:
We pick w
i
o
t
/'
/'
[\bn(W)_bCD(,-,)1 s s
+ \cn(w)_BCD(w)/ s s
+
and a
O.
0.65) implies that there exists an
Consider an infinite subset ::tPc]N. infilldte SUbset ]Nil (w)cJN'
-N:(w,h)IJdFs(GJ)
dFs(w)-full subset
D(w)cJR+,
with
0.66) An j\oo, N ("",h) = N_,... ,h)
s
In particular.
lim
An
n
ACD
....,,() M ('"',h) = N (w,h ), w
S
0
S
s
which yields:
0
lim a 1 CD \z l>aJ) = O. Hence. since Jrl' is dense in C, it is easy to check that 0.66) holds for all h EC, and in particular h::lfa (recall that f(z)=z for z small enough ) ; Then 0.64) and 0.66) imply that: limn .., -.rll() Hn(w)wn''U(LJ) = HCD(w). in by (3.70), """'n.""oo m Z _ _Z in • A simple computation shows that the predictable
process with finite variation in the canonical decomposition of
BD.,
while
I t,Zn I,; 1.
because of
\f I,; 1..
Zn
is
Thus (3.72) implies (3.20,11).
By (3.71) we have V([Zn,j,zn,k J
(3.76) Since
Cn,jk
- [ZOO,j,zoo,kJ \
0,
h(z)!( Izj21\1.)
0.
For each
h: lRm----'P:ffi
•
it follows that
continuous, such that
L .Sn AA S
S
S
lH (2 < Y_ I f I > +
is a local martingale. Thus
If' /2..iA +
L
j .. d
crt j j
G ••
210 IV
A/,."
Proof of (4.12). By (2.26,a), it suffices to prove that if is a good extension of (Q,r,I,p) and if X and X' are two solutionprocesses on this extension, then X= X' Pa.s. Let Y =X X' , cT'(n) = inf(t: IXt \ '" Il or n), g = g(5b g(XI). Then we have Y = goZ •
'Xt
Since of Z on this space ded by (4.1), Y is tion is Y = M I + F' , to this process Y,
is a good extension, the canonical decomposition is still Z = M+ F, and since g is locally bouna special semimartingale whose canonical decomposiwith M' = geM and F' = geF. We will apply (4.13) and in (4.14) we have .,, jj
fl
=
L
",jk(1"kl "'jl k,l..:m g g.
Let nlOE be fixed, and G (depending on n ) be given by (4.16). We have that N = Iy 12/ H G is a local martingale. But (4.16), (4.17), .-.J tV ,v ,v the facts that g=g(X) g(X') and that Y=XX', and (4.10), imply that Gt,,;;; 0 i f t c:r(n). Since Iy 12/ H :;0 0, i t follows that ,.J
(Ntt\o(n»t"O is a local martingale that is nonnegative. Hence it is a nonnega ti ve supermartingale, and since NO = 0 we must have Nt Ao(n) = 0 Pa.s. for all t .. O. Hence ]yj2/H = G on [O, 2(2d+l).
I
x
y}.
It holds
By Holder's inequality,
E!nt(x,y) - nt,(x' ,y') I
p
+
-
1
2p
1
) 2} .
221
By Lemma 1.3 and Proposition 1.1, we have
(1.7)
Elnt(x,y) - n ,(x',Y')1 t
-p
< Cp, T IX-YI
-
-2
< c8
p,T
if
P
!x-y!
2.
and
8
-p
p
p{ Ix-x I
P
I
Ix'-y'l
+ Is-s'
P
I
2. 8}.
Since
we get the assertion.
T
+ Ir-s' I +
2) t- t
%
'I }
.J2.
21 t- t I2 }
+
2. 8, where
Then by Kolmogorov's theorem, nt(x,y) {(x,y)! Ix-y!
p
Ix' -y' I {Ix-x' I
I
is a positive constant.
C
p,T
is continuous in
[O,T] x
0 are arbitrary positive numbers,
and
The proof is complete.
The above lemma leads immediately to the "one to one" property of the map
for all
t
a.s.
We shall next consider the onto property. Lemma 1.5.
Let
is a positive constant
T >
a
and
p
We first establish
be any real number.
Then there
such that
1f t
(1. 8)
Proof.
We shall apply Ito's formula to the function
t
(x»
- f(x)
= L
i ,j
Jt
0
zi
s
J
s
(x»dM
j
s
E
[O,T].
222
Let
K be a positive constant such that I
I
J
holds for all
i
< K(l +
and
j.
Ixi
2
)2
Then,
Therefore,
IE1tl < 2rldjp!KfE(I+ It: s(x)1
2)Pds.
o
Similarly,
so that
IEJ t
I
Therefore we have
< Iplrtd + 2Ip- I
I)K2f E(1 o
+ It:s(x)
12) Pds .
223 E(l + !!;t(x)1 2)p
2.
(1
+ Ixl
+ const.J\(l + !!;s(x)1 2)Pds.
2)p
o
By Gronwall's inequality, we get the inequality of the lemma.
Remark. Therefore, inequality (1.8) implies
(1. 9)
Now taking negative
P
in the above lemma, we see that
tends to infinity in probability as
x
We shall prove a stronger convergence.
Lemma 1.6. of
Rd.
r:
nt(x,w)
We claim
be the one point compactification
Proof.
1
if
is a continuous map from Obviously
nt(x)
x
= 00
[0,00) x
is continuous in
into
l R
a.s.
[0,00) x Rd.
Hence it is enough to prove the continuity in the neighborhood of infinity.
Suppose
p > 2 (2d + 1) •
I
tends sequencially to infinity.
Set
nt.(x)
Then
d R u {co}
Let
!!;t(X)
It holds
224
By Holder's inequality, Proposition 1.1 and Lemma 1.5, we have I
I
Eln (x) - n (y)I P < (En (x)4p)4(En t s t s
t
(x) _
I
s
(y)1 2p)2 £.
Cp,T(l + Ix/)-P(l + lyl)-p(lx-yIP + It-sI
if
Set
t,s
E
[O,T]
and
x,y
-x1 = (xl-1 , ... ,x-1 ). d
E
d
R , where
C
p,T
is a positive constant.
Since
we get the inequality
Eln (x) - n (y)I P < C (1 1 _ liP + It-sI t s - p,T x Y
a
2).
Define
n (x)
if
x., °
if
x
t
=
°
Then the above inequality implies £.
E/n (x) - n (y)/P < C T(lx-yIP + It-sI t s P,
In case
y = 0, we have
2),
x., 0, y
°
2)
225
Therefore
nt(x)
This proves that
d
[0,00) x R
is continuous in nt(x)
is continuous in
by Kolmogorov's
[0,00) x
theorem.
neighborhood
of
infinity.
Proof of "onto" property (Varadhan). t
= Rd u {oo}
on
Define a stochastic process
by
A
E;t(X) if
Then
t
(x)
Thus for each map on
is continuous in
[0,00) x
t > 0, the map
t(o,w)
Rd
by the previous lemma.
is homotopic to the identity Then
which is homeomorphic to d-dimensional sphere
t(o,w)
is an onto map of
Now the map
t' we see that
by a well known homotopic theory.
is a homeomorphism of
one, onto and continuous. map
x
E;t
Since
00
Ad
R , since it is one to
is the invariant point of the
is a homeomorphism of
Rd.
This completes
the proof of Theorem 1.2.
2.
Smoothness of the solution In the previous section, we have seen that the solution
of Ito SDE (1.1) is a homeomorphism for all ficients are Lipschitz continuous.
t;t(o,w)
t, provided that coef-
We shall show in this section that
226 k-c1ass, if coefficients are of C then for any
t.
is a Ck-1_diffeomorphism
We first state
Proposition 2.1. that coefficients
and Freid1in
X ••• ,X 1, r
t
a.s.
I
where
I
Proof.
in
k=l
Jt
X_'
0 --k
s
(x)
s
(x)
satisfies
cUt,s
(--) dX j
Following [2], we will give the proof.
(0, ... ,0,1,0, .•. 0)
(1
d
+ he£, where
R
and let
x' = x
Then it satisfies
(2.2)
+ r2:
(.::.::.t.) dX. J
is the identity matrix and
Xk(x)
e£
k-1_c1ass is of C
Then
Furthermore Jacobian matrix
(2.1)
Suppose
k-c1ass of equation (1.1) are of C and
their derivatives are all bounded. for any
12]).
11 (x,x') t
is the £-th
h
component)
Let
be a unit vector
is a non zero number.
Set
227
S (x,x') = (x), (x'), n (x,x')) is a t t· t t 3d-valued R stochastic process with Lipschitz continuous coefficients.
We may consider that
Then by Proposition 1.1, we have
E!St(X'x') - ss(y,y')jP
Therefore
St(x,x')
2
is continuous in
dX£
exists and is continuous in
To get (2.1), make
t
(2.3)
h
(t,x,x')
in
2d• [O,T] x R
and has a continuous extension to ---(x)
+ jx'-y'jP + It-sI
tend to
(t,x)
°
122)
[O,T] x
This proves that
for almost all
in (2.2).
w.
Then we obtain
(x)
This proves (2.1). Consider next the SDE for the pair
Coefficients
dX£ (x)).
of these equations are of Ck-l-class and their derivatives are bounded. We may apply the same argument to the pair. have continuous derivatives
Then we see that
provided
k > 3.
---
dX£
Repeating this
argument, we get the assertion of the proposition.
The smoothness of coefficients
X j
is a local property and boundedness of
and their derivatives are not needed.
boundedness is not satisfied, choose for each class functions xjn) , n=1,2 •.• and derivatives of each associated to
xi
n)
X(n) j
, ... ,x;n)
such that
X j
xjn) (x)
are bounded. coincides with
In fact i f the
a sequence of Xj(x)
if
Then the solution for
Ixl < n i;(n) (x,w) t
228
t
0
k-l C -class in
II; (x,w) t
Ixl .::. m < n
infinity, we see that
if
T
n
I ->
n },
Therefore
(x,w) > t.
Since
is of
I;t(x) T
n
(x)
tend to
k-l is of C -class everywhere for any
I;t(x,w)
It remains to prove the smoothness of the inverse map
-1
t ,
We
I;t •
claim Lemma 2.2. (c.f. Ikeda-Watanabe [8]) t > 0
singular for all Proof.
x
Dl;t(x) is non-
a.s.
We shall consider a matrix valued SDE for each r
(2.4)
I -
The solution
and
Matrix
K t
t
K X-' (I; S-K s k=l Jo
L:
satisfies
k s
(x) dM
K DI; ex) t
x
t
I
for any
t.
In fact, by
Ito's formula we have
(2.5)
dK °DI; (x) + K dDI; ex) t t t t
Substitute (2.1) and (2.4) to (2.5), then we see that the right hand side of the above is
O. This proves
KtDl;t(x)
=
I, showing that
Dl;t(x)
is nonsingular. Now the inverse mapping theorem states that k-l a C -class map.
Theorem 2.3.
We have thus obtained the following theorem.
Suppose that coefficients of equation (1.1) are
of Ck-class and their first derivatives are bounded. k-l is a diffeomorphism of C -class for any
Then the solution t
a.s.
229
In later discussion, we shall mainly concerned with Stratonovich SDE.
Therefore it is convenient to get analogous results for Stratonovich
SDE.
Let us consider a SDE
(2.6)
t
where the right hand side denotes the Stratonovich integral.
If
2 are of C -class. the equation is written as the Ito SDE
(2.7)
Therefore if
Xl' ...• x r
together with their first and second derivatives
are bounded, all coefficients of equation (2.7) are Lipschitz continuous. Then the solution
is a homeomorphism for any
t
a.s.
We then
have
Theorem 2.4.
Suppose that coefficients
SDE (2,4) are of Ck-class (k
2).
XI, ... ,X
of Stratonovich
r
Suppose further that
XI, ••• ,Xr
together with their first and second derivatives are all bounded. the solution
k- 2-diffeomorphism is a C for any
t
Then
a.s.
Here,
Let
M be
CO-diffeomorphism means a homeomorphism.
3.
Case of Manifold In this section, we shall consider SDE's on manifolds.
a a-compact, connected Coo-manifold of dimension Ck-vector fields on
M where
k >
2
and
d.
Let
XI' ••••Xr
be continuous
be
230 semimartingales.
We shall consider SDE on the manifold M;
(3.1)
A
o
since
1
be a geodesic
II
(_a)
a l Yo
The covariant derivative
y(t)
exp
x
o
(3.6)
II
-1
increases the distance. YO such that y(O) = YO and
(_a)
al Yo
X at
YO
a
II Since
exp
is written as
a
i
(-k) II z x k(YO) (-.) ay Yo i ' ay1. Yo
increases the distance, we have
y(t)
X(yO)
II.::. II
a
.
2
(k)y II (l: xl.k (yo) ) 0 i ' ay
1
2
1
2 2 .::. (l: x k(YO) ) i
i
'
Note that
a
i
x and YO'
a
k ax
k(YO) = - k , ay
k(YO)
=
0
Then we have i
x (yo)
Remark.
i
x
since i
X,k(Y O)
is bounded in
(yo) - l: j
(y 1 ,
.
J
d ,Y)
.
k(YO)XJ(y o)
is a normal coordinate with origin
= k a x i (yo)' ax
Inequality (3.6) implies that
Yo'
If the sectional curvature of a connected complete
Riemannian manifold is greater than a positive number, then the manifold is compact.
Hence the solution of (3.1) is always a flow of diffeomorphisms
236 We shall finally consider the equation (3.1) when the Lie algebra generated by vector fields
Xl ••..• X r
is of finite dimension.
We will
not assume any condition to the manifold where the equation (3.1) is defined. For two vector fields XY - YX.
X. Y. we define the Lie bracket
It is again a vector field.
fields
il •.•. ,i
n
n l
it as
as
The Lie algebra generated by vector
tx.1.
is the linear span of vector fields
[X.1. _ ,X.] ••. ]. n=1,2, ...• where 1.
[X.Y]
E
n
1
[X. [ •.. 1.
{1,2, •.. ,r}.
2
We denote
L.
Theorem 3.8.
Suppose that
Xl •...• X r
00
are complete C -vector
fields and that the Lie algebra generated by them is of finite dimension. 00
Then the solution
of SDE (3.1) is conservative and is a C -
diffeomorphism of
Proof.
M for any
t > 0
a.s.
We need a fact from differential geometry.
(e.g. Palais [17]) that any element of exists a Lie group
product manifold
of
M. i.e. there esists a C -map
there exists
M such that (a) for each
from the g ¢(g,o)
is
M and (b) ¢(e.o) = identity, ¢(gh,o) = ¢(g.¢(h,o)) G.
(Lf.) The map
g --..¢(g, 0)
is an isomorphism from
G (= right invaraiant vector fields). X of
(L) G is a Lie ¢
G into the group of all diffeomorphisrns of M. (iii) Let Lie algebra of
It is known
is complete and that there
00
G x Minto
a diffeomorphism of g, h
L
G with properties (L) - (iii) below:
transformation group of
for any
w.
such that
be the
For any
X of
L
237
(3.7)
X(f o ¢ ) (g)
Xf(¢(g,x»
x
00
holds for any C -function
G such that Now let
X.J
(j=l, ..• ,r)
Here
fo¢x
00
is a C -function on
LX.J
t
relating to
be elements of
Consider SDE on
j
X. J
by
G
) odMjt
is a Brownian motion, the solution
If
a Brownian motion on Lie group [9] •
M.
on
f o¢ (g) = fo¢(g,x). x
the formula (3.7).
(3.8)
f
G.
is so called
Ito has shown that
it is conservative
His argument can be applied to the above (3.8), provided that
j=l, ••• ,r
satisfies property (1.4).
general
j=l, ... ,r
Then the conservativeness for
can be proved by the method of time change, as
we have stated in Section 1. Set each
(t,w),
=
where
is the unit of
is a diffeomorphsim.
fo¢(e,x) + L j
f(x) + L j
Therefore
e
I
I
We have
t
X.(fo¢
o
J
x
t
o
J
is a solution of (3.1).
s
G.
(x»odM
s
(e»OdM
j s
j s
The proof is complete.
Then for
238
4.
Decomposition of solutions Consider a Stratonovich SDE on a manifold
M:
r
(4.1)
j=l We shall assume from now that vector fields for simplicity.
The solution 1
Xl"",X
r
are of
00
C
is a functional of vector fields r
Ms, ••• ,M O 2 s < t, obviously. s' how the functional is written explicitly.
We are interested
We begins with a simple case.
The following proposition is more or less known.
Proposition 4.1.
Suppose that
fields and commutative each other.
Xl""'X
r
are complete vector
Then the solution of (4.1) is rep-
resented as
(4.2)
where
Exp sX
generated by
Proof. f
(x
_00
i,
X., 1
< s
t}.
the differential from
as
Dt(W)
onto
Rt(W).
Rt(W).
is a x
of
Dt(W) ,
is defined as a linear map
such that
X
x
t
IrJ X
)
x
E
T (M).
x
Given a vector field
X
at the point
We define a new vector field
x
Then
Given a point
of the map
to
Tx(M)
T(X,W).
It is the domain of the map
Denote the range of the map diffeomorphism from
(4.1) with life time
E
M.
on
M, we denote by
-1
(x)
X -1
(x)
X
x
the restriction of on
R t
X
by
'
Then it holds
00
for any C -function Let
1
f
d
(x , .•• ,x )
on
M.
be a local coordinate.
we see that the i-th component of is
i
(X) (x)
Taking
f(x)
= xi
above,
ralative to the coordinate
241
(__d_ dxk
Hence, denoting Jacobian matrix with components
-1
Now let
as
t
the vector
is
be the inverse of
The vector field
Then it holds
-1
(X) f (x)
for any COO-function f on M.
With a local coordinate
1
d
(x , ... ,x ),
we have
-1
Remark. and
If X
and 5.3
X
hold.
is commuting to all
X
then
These properties follow from Proposition 5.2
of the next section.
Suppose now we are given
other
M and continuous semimartingales
(4.4)
Xl"",X r'
00
C -vector fields
Yl""'Y s
Consider SDE
on
242 on
A sample continuous stochastic process is called a solution of (4.4) if
o(x)
t < o(x)
M with life time
is in
D t
for all
and satisfies
f(s (x» t
00
for all C -function on
M.
Then we have
We shall first ohtain SDE governing the composition map
Proposition 4.2. t
E
[O,o(x»
satisfies SDE
(4.5)
Proof. f
We shall apply an extended Ito's formula [14].
00
be a C -function on
coordinate
1
Ft(x) =
M and let
d
(x , ... ,x ), we shall write
L
j
J
t
(x»odM
we have by Theorem 1.2 of [14],
j
t
St
t
as
(x).
Let
Using a local
1 d (St"",St)'
Since
243
(4.6)
L
dFt(l;;t(X»
.
. aFt 01;; (x»odMJ + L ---.(1;; t t t. t
J
J
aX
. t
The second term of the right hand side equals
L
i,k where
is the i-th component of the vector field
1 d (x , ... ,x).
relative to the local coordinate
L
k
t
t
L
t
k
The above is equal to
k )(1;; (x»odN t t
01;; (x»odN t
k t
Hence (4.6) is written as
df(nt(x»
The proof is complete.
Remark.
Instead of (4.4), consider
L Yk(K )odNk .
(4.7)
k
Then the composition
(4.8)
t
At -
j L Xj(A )odM t t j
t
t
oK t
+L k
satisfies the equation
t
k * (Y )odN k)(A t t
244 This can be proved analogously as Proposition 4.2. We can now get the decomposition of solution of (4.1)
Theorem 4.3.
Consider two SDE's r
l: Y (I:; ) od}1j j=l j t t
(4.9)
(4.10)
If
Xj = Yj + Zj' j=l, •.. ,r
o
0
a.s.
The second half will be obvious.
Corollary. n
(x)
and the
n (x) t
is the solution of (4.1) for all
t
The first half of the theorem is immediate from
Proposition 4.2.
then
Furthermore, i f both of
are flows of diffeomorphisms, then so is
n (x) t
= I:;tOnt(x),
hold, the composition
If
Zj' j=l, ... ,r
are commutative to all
Yl""'Y r
of the theorem is determined by r
(4.11)
Proof.
.
l: z . (n ) odMJ j=l ] t t
Since
Zk
by Proposition
5.2, which will be established at the next section.
A typical example of the decomposition of the solution is that of linear SDE on
d; R
245
where
A is a dXd-matrix, B
is a dXr-matrix and
Wiener process.
The equation is decomposed to
Clearly we have
s. t (x)
e
At
x.
: e
Consequently, nt(x)
i;t (x)
x +
-At
W t
is a r-dimensional
Then
B.
t
Jo e-AsBdW.s
We have thus the decomposition
eAt(x + Jt e-AsBdW s)' o
Some other examples of decompositions are found in [13].
We will mention that the technique of the decomposition is used in filtering theory in order to get a "robust" solution (c.f. Doss [6], Clark [4] and Davis [5]).
where
Consider a SDE on
is a Wiener process.
d R
Suppose that
are commuting each other, but they are not commuting with the equation to
XO'
Decompose
246 r
t
= L
j=l
= Exp W1X
Then. it holds
t
Then
1
0
and its Jacobian matrix
are locally Lipschitz
continuous with respect to the Wiener process and
if
N > 0
there is a positive constant
II w. (w) IIT
N
II w. (;;:;)
and
Consequently the vector field continuous with respect to
II
W. (w)
-1
IIT =
sup Iw (w) O = A(f)a It is obvious that second order forms can be multiplied by COO functions on M. So we may also define : DEFINITION. Let f and g be COO functions on M. Then df.dg is the second order differential form 1 d 2 (fg)-fd 2 g-gd 2f ). df.dg = 2( We know that T(M)CT(M), so each second order form has a restriction to T(M), which is an ordinary form. It is obvious that d2fIT(M) = df df.dgIT(M) = 0 Remember now that we have global coordinates on M. We prove the intrinsic character of the "full second order differential of f" as it appears in
262
the cla£sical books THEOREM (trivial).
d 2f = D.f d 2xi + D.. f dxi.dx j 1
lJ
Proof: fix aeM. We don't change anything by replacing x i byi x i -a , so . . . 1 2 . . J= J we may assume that x1(a)=O, in which case dx1.dx (x1x ) , and the for2fl 2g1 mula reduces to the fact that d = d ,where a a g(x) = Dif(a)x
i
1
i
+ 2Dijf(a)x x
j
This is just the Taylor formula of order 2 : f(x)-f(a)-g(x) has a zero of order ?3 at a, so all differential operators of order 2 at a vanish on it. Just out of curiosity, it is natural to wonder about forms of higher orders. It turns out that they exist, that one can define the d and • operations in a nice way, but there are deep differences between orders 2 : essentially, for n>2, they are dual to something larger than differential operators of order n. They seem to be quite useless and inoffensive.
Our next step consists in extending the product • and the differentiation d to arbitrary 1-forms. This is now obvious. Given two 1-forms p=a.dx i 1 and a=b.dx j , define J i j 2i j i p.a = aibjdx .dx dp = aid x + Djaidx .dx We must check that the result doesn't depend on the coordinate system. Now we have the properties : - the product is commutative, and bilinear w.r.to Cill (M) multiplication - d(fp) = fdp+ df.p which in turn characterize both operations uniquely. Note that (18)
dpIT(v) = p
p.aIT(v) = 0 •
There is another intrinsic characterization of the d operator ( which doesn't extend to higher orders ). Let w be a form of order 1, and h(t) be a curve in M. Then we have d
•
-
h(t),w> = < h(t), dw> •
Differentiation raises by one unit the order of forms. There is another such operation, deduced from a linear connection f • Since f maps linearly T (M) into T (M), its dual ( which we denote by f too) maps a a forms of order 1 into forms of order 2, with the property that (19) f(p)IT(V) = p r(fp)= ff(p) • The Christoffel symbols appear in the expression of f (20) r(dx k) = d 2xk +
as follows
lJ
Note that d-f is a second order form whose restriction to T(M) is zero, i.e. just a symmetric bilinear form. If w = aidxi is a form
263 k
.
.
(d-r)w = (Dja iand on a Riemannian manifold we may take the trace of this quadratic form w.r. to the metric to get the scalar function k ij (21) -ow ( just consider the left side as the definition of 0 on forms, since we don I t need any general theory of the 0 operator, and remark that -odf = according to (10)).
f:::,f
6. ITO AND STRATONOVICH INTEGRALS In the usual set-up of stochastic integration, the Ito integral has an awkward geometric status ( it doesn't"behave well"under a change of coordinates ), while the Stratonovich integral has an awkward analytic status ( as Yor [1J shows, the approximation procedures which are traditionnally used to justify its use aren't valid for all semimartingales). The use of second order forms will clarify the situation. Roughly stated, the true stochastic integral is a second order object, like the semimartingale differentials themselves. To reduce it to first order, one may use two geometric procedures, which yield Ito or Stratonovich integrals. Also, remember that forms zhouldn't be integrated only on paths, but on chains, i.e. on paths provided with formal multipliers. Here our multipliers will be predictable ( locally )bounded processes. For simplicity, we shall omit the multipliers most of the time. a semimartingale with values in M, and let Q
=
a i d x l +a i j d x l .dx J be a (Coo) form of order 2 on M. Then we define the stot chastic integral /t Q X
Yt =
/t Q Xo
=
/t
o
l
S
of Q along the path X o i )dX + s
as the real valued process
Einstein convention
Let (Kt) be predictable process. Then the stochastic integral of Q alors the chain KoXt is the process --0 t t . 1 t . . / Q = / K dY = / K a. (X )dX l + -2/ K a .. (X )d KoXt 0 s s 0 S l S S 0 S l J s s o
PROPERTIES. 1) Probabilistic. Those of the usual stochastic integrals in : stochastic integrals are real valued semimartingales ; they remain unchanged if P is replaced by an equivalent law Q ( more generally, if Q«P the P-s .i. is a version of the Q-s.i. ) ; they are local on 0 ( if two semimartingales X and X' have the same path on some subset A of 0, the corresponding s.i. have the same paths on A ). Etc •.• The use of mUltipliers is convenient at many places. For instance, if U is a coordinate patch, it is convenient to use the multiplier I!XeU! to localize. If S,T are stopping one uses the multiplier IJS,TJ'"
264
2) Differential geometric. The first main property, of course, is the fact that it is intrinsic ( this is another expression of the principle of Schwartz: incidently, the principle of Schwartz itself might be recalled by a notation like It K d 2X ,G > ). More generally, let F : M N be o s s a map, on let G be a form of order 2 on N. Let also Z be the semimartingale FoX with values in N. Then we have It G It F*(G) Z X o 0 Also note the following simple formulas (22) I d 2f
the pull-back of G on M ).
Xto
It df .dg =
X
f(X), g(X) >t
o
DEFINITION. Let w be a ( Coo) form of order 1 on M. Then we define its (Stratonovich ) integral along the chain KoX as
I
(23)
I t dw KoX
t w o
KoX
o
Assume a linear connection f is given on M. Then the Ito integral of w along the chain KoX is (24)
(I)
I
KoX
tW o
I
t
KoX
fw
o
Let us pause for a discussion, since these definitions are the main point in this report here, Ito and Stratonovich integrals are given the status, both oan be used with arbitrary predictable multipliers. ( 80 the usual statement that S-integrals are less general than I-integrals is no longer valid here : this is due to the fact that we are working with COO forms. For qUite general forms the d operation would require more regularity than the r operation ). The Ito integral requires more structure than the S-integral, which is the main geometric object, as was discovered by Ito himself, and confirmed by all the subsequent work on the sUbject. Finally, we remark that no approximation procedure, no smoothing of the path, has been used to define the S-integral. SOME PROPERTIES. a) The main property of the Stratonovich integral, as noted by many authors ( personnally I learnt it in Yor [1J ) is the following: if w is a closed form, then It w is just the integral of w, in ---
X o
the differential geometric along the continuous path For an exact form w=df, this reduces to (22), the general case requiring a localization. b) Let F : M---;;oN be a map, and w be a form on N, Zt be
Then
265
we have
It F*(w) • This corresponds to the second order formula
X
0 * * just before (22), and the obvious property that F (dw)=dF (w). The corresponding relation for Ito integrals is a rare event. Indeed, denoting by the * (rw) same letter r two connections on M and N, the property that r(F* (w))=F is extremely restrictive. For reference below, note the formula on N , F*(rw) = r(F*(w)) +
(25) If
where the "greek" coordinates xC>' refer to N • c) The main property of Ito integrals is their relation to martingales. X is a martingale with values in M ( relative to r ) if and only if = (r)/ t w is a real valued local martingale for any form w of order 1. In X thisocontext, trivial identity (26 )
It w
X
o
(r)/ t w + It (d-r)w X
0
X
0
appears as the true expression of Ito's formula in a manifold M, since 1) it reduces to it when M=Rn with its trivial connection, and w=df, and 2) if X is a martingale with values in M, it gives the decomposition of the left side in its local martingale and finite variation parts. Let us give two applications of these computations to the Brownian j motion X of a Riemannian manifold M. In this case, if g=aijdxi.dX is a second order form reduced to its quadratic part, we have 1 t i' 1 ij (27) It g '2 I a .. (X )d = -2G(X )ds, where G = a .. g 0 s s s X o
1) Applying this to formula (26), and taking formula (21) into account, we get that G = -6w , a nice formula due to Ikeda-Manabe [1J. 2) Let us return to the situation of b), and look for the condition that F(X t) be a martingale with values in N • Looking at the rigbh side of (25), the first term gives a martingale by integration, while the secend is the purely quadratic second order form Q Y D .. F Q -rk.. Dk F Ct a f Ct .. d x i .dx j f.Q . + r eyo F Di FeD j F
Applying (27), we see that the condition is the vanishing of the functions Q G = .gi j • This is exactly the definition of a ha-rmonic mapping F : JVl---Y N ( Hamilton [1J, p.4 ). d) Let us end this section with a basic property of the Stratonovich integral. Let w be a form of order 1, and let f be a COO function. If we know the real valued semimartingale Y It w , then we may compute the t= X integral It fw = Zt by ordinary Stratonovi8h integration. More generally, if K is a (28)
X
0
predictable (locally)bounded process
I.Xt
o
fw
ItK f(X hdY 0 s s s
jtK f(X )dY + s s s
0
Y>s
266
7.
SEMlMARTINGALES AND PFAFF SYSTEMS Consider a distribution of submanifolds of dimension p in M, described as 1 ( stable under multiplication by cro functions ) of on the distribution. Locally we may describe
usual by the space ro all c forms which
..• ,n-p).
the distribution by the vanishing of forms avoid localization difficulties, we assume that the
Since we want to
are independent at
each point and describe the distribution globally. A differentiable curve h(t) then is an integral curve of the distribution if and only if
It
h
= 0
for
1,2, ••• ,n-p
o It is entirely natural to say that X is an integral semimartingale for the distribution if we have YCt = I t wCi = 0 for 1,2, ••• ,n-p • t Xo This property doesn't depend on the choice of the basis wet • Indeed, let w be any other form that vanishes on the distribution, aud let
Yt=/ t w ;
X Y = Itg (X O. On the other hand, 0 the t Oi OOiS s geometric meaning of (29) isn't at all obvious, except in the trivial case
writing w=g
=
we have
of a completely integrable system : then we may assume that (locally) exact forms, and moves in some
We
write
(29)
(29)
are
simply means that the semimartingale
=
Ci
integral manifold F
constant (
Ci
=1, •• ,n-p).
in its explicit second order expression:
(0)
It dw = 0 for X o and remark that weJ => fwe1-, and so
we.:!: It df.w X o
=
I xt0
d(fw)-fdw = 0 • Using the
Schwartz principle in the reverse direction, we may say that a second order tangent vector
L
belongs to the distribution if < L,dw > = 0 for all weJ
( therefore, = 0 for any form G ). It turns out that any distribution has some non trivial second order integral fields: namely, if A and B
are first order integral fields, then AB+BA is a second order integral
field, thanks to the formulas
(1) (32)
< AB, dw > = A -
>
0, exterior differential)
< AB+BA, dw > = A + B •
So AB and BA are second order integral fields if and only if AB-BA is a ( first order) integral field. 8. THE LIFTING OF A SEMIMARTINGALE THROUGH A CONNECTION The results on this section were explained to me by Schwartz. They are generalizations to general connections and general semimartingales of the classical "stochastic parallel displacement" theory, due to Ito and Dynkin.
267
The extension to general connections can be found also in Malliavin [1J, for brownian semimartingales. The geometric "second order language" can possibly bring some additional clarity to the subject. We shall use the "horizontal subspace"point of view for connections. For simplicity, instead of considering a fiber space, we consider just a product W=UxM, with global coordinates (xi) on the "base" M (19:::n) and (xO') on the "fiber" U As usual, 11 denotes the projection on the base, but we mention it as little as we can: if g is a function on M, we also denote by g the function gol1 on W • This concerns in particular the coordinates xi, and D. has a double meaning, as %x i on M and %x i on W. l
A connection r is a distribution of subspaces Hx ,u e Tx ,u(W), called horizontal subspaces, such that V(x,u)eW , 11* is an isomorphism of Hx,u onto - Tx (M) so Hx,u is supplementary to Vx,u =Ker(I1*1 x,u ), the vertical subspace of Tx,u (W». Then any tangent vector teT x (M) has an unique horizontal lift H(t) at (x,u)eW • To compute H(t) it is sufficient to know ( often denoted by V.l )
(33)
Going to the preceding section, we see that the distribution of horizontal subspaces is associated to the forms (34) gO' = duO' + l
Then any semimartingale (X on the "base" M has a unique lift Xt t) values satisfies the Stratonovich with prescribed initial gO' 0 differential equations that is Xo ) i (35 ) dU0't + r Ct( i Xt,Ut *dXt = 0 of course, even if r is CCO , one must be careful about the possibility of an explosion in (35). and X may have a finite lifetime. t We want to compute the second order tangent vector to the lifted semimartingale X , that Ls 2
xj D d _vi uO' D ' > t ij + and to compute from (36) the brackets .
(R 3) For any X,Ys*(M) (infinitely differentiable vector fields)
is an infinitely differentiable
function.
A Riemannian connection is a map
*
(H)
x
*
(1.1)
satisfying the following conditions:
(Co)
Xll
of independent random variables with
the cornmon distribution Pie Edt} = e-tdt
(3.2)
n A sequence ( x ( n) , c,c ( n ) , n ( n ) ' •••••• , n ( n )) E T ( k+ 1 ) (H) . 1 k
defined inductively as follows x(o) = x , £;(0) = £;, niO) = nl"" y (n)
x
(n+1)
,£;
-
nj
(n=O,l, ... )
(n)(e n+ 1) (n)
n(e n+ 1;n j
...
)
)
(n=O,l, ... ) where llx(dE;) is the uniform law on
Le. the unique prob-
ability law which is invariant under the orthogonal group defined by
< > .
means of the inner product
The isotropic transport process is defined by x(t)
Y
£;(t)
Y(x(n) ,£;(n)) (t-1 n )
I] .
J
(x
(n)
(t) = 11(t-1
, £;
(n) (t-1 n) ) (
"
("
n;n;n))
"
where 1
= e ... +e It can be shown that this defines a Markov n 1+ n. jump process on T(k+1)(M). Let at
00
e
be the space of differentiable functions on M which vanish
In order to compute the infinitesimal generator of the iso-
tropic transport process, we introduce the following operators: Pf(x,n 1,···,nk ) ,!, :
f(x,£;,n
x
1,
.. ·n ) lJ k x(dE;)
(fEe, t.>0)
277
"'_At
- Ie
o
f(y(t),Y(t),n
1(t),
... ,n
k(t»dt
(fEC, A>O)
(fEC,t>O)
(fEC,t>O)
It will be clear from what follows that these operators map C into C. t I[Zk+P-I]T f ds o s
=
t IT (Zk+P-I)f ds, 0 s
fEC
To prove this, it suffices to obtain the corresponding result for Laplace transforms.
To prove this, we first obtain
= f = k) From the smooth dependence on initial conditions,
Lemma 3.1.
into C.
maps C into C and (A-Z
The result now follows from Laplace transformation of
Propos i t ion 2.2. Lemma 3.2. Proof.
maps C
RAf '1
E{I o
+
I }e -At
.•.•• ,nk(t»
'1
The first integral is
'"
Ie
o
-t -At e f(y(t),Y(t),n1(t)' •.•..•
The second integral is
dt
278 E {J e -At f (x (t ) , I;(t ) ,11 (t ) , ..... , 11 (t ) ) d t } k 1 T
E{e
E{e
-AT 1
-hI
00
EJe
-AS
o
00
EJe
°
-AS
1
f(X(Tl+s),I;(Tl+s),1I1(Tl+s), ... ,lIk(Tl+s))ds}
f(x(s;x
(1)
),I;(s;1;
(1)
),1I
(1) (1) ), ... ,lI k(S;lI )ds} 1(S;1I1 k
Lemma 3.2 immediately implies the series representation
From this it may be shown that R
A
maps C into C, in particular RAf is
in the domain of Zk'
Lemma 3.3. Proof.
(A-Zk -P+I)R f = f A
Apply (I+A-Z
k)
to Lemma 3.2 and use Lemma 3.1.
Thus we have proved that RAf-A
-1
f
=
1
A- (Zk+P-I)RAf
By inversion of
the Laplace transform the first part of the theorem is proved. prove the second part i t suffices to show that ¢ for any fEC.
R (A-Zk-P+I)f-f A
Clearly 0) t
-
Brownian motion, then, for any (L;«)
1 v2 E[exp(- -2
Jto
is the radial part of a two dimensional v >0 : ds/R 2) IR .. a, R .. yJ .. I (ay/t) / I (ay/t) sot \I 0
That is to say, the first Hartman law with parameter tional law of ds/R 2 given R .. a and R .. y. o s o t
f
ay/t
is the condi-
The key to (l.e), and apparently to most results concerning ratios of Bessel functions as completely monotone (c.m.) functions of
IV ,
is a suitable
Cameron - Martin - Girsanov type result which relates one Bessel distribution to another. One advantage of this approach is that many ratios of Bessel functions in Ir and in variables.
I\I(r)
and
KV(r), known from the literature to be c.m.
IV separately, turn out to be jointly c.m. in these
287
The representation (l.e) of the first Hortman law is
disappointing
in several respects. It involves a conditioned process (or Bessel which is an inhomogeneous diffusion), and as a result the connection with the other Hartman laws (l.e) is so obscured that it is not clear why the law is infinitely divisible. Also, the second Hartman law is still more deeply hidden in the Bessel bridge. These matters are rectified with the help of the two parameter family of Bessel diffusions drift
BES(ll,o)
of Watanabe
[46J,
with index
u
0
and
o.
Our justification for the term "drift" is the result of d Rogers and Pitman [4cD that if X is a BM in R with a drift vector of 0 I) then X is a BES(ll,O) with index j J " (d-2)/2. In section 4 we present BES(ll, 0
(l.g)
.. inf{t : R .. x}. t
be the first time that the radial motion hits (l.h)
P with Xo .. Yo
E exp [ - 2I v 2
fOOT
x
dS/R
21
sJ
x. We show that
.. I ( 0) t
makes the infinite
-
divisibility of the first Hartman law plain. by decomposition of the
T for y > x. and the laws with the Laplace transforms (l.e)
integral at appear
y
as factors. Next. time reversal reveals a'tiual" representation for
the second Hartman law. namely E exp [-
i
v
2
J:
J ..
dS!R 2 s
x
where
L
sup{t : R t
x
= x}
K (ox) ! K (ox) V
0
is the last time at
x.
In view of the last exit decomposition of Pittenger and Shih
[39]
[j,6]. Hilliams [49J). the infinite divisibility
(see also Getoor and Sharpe
of the second Hartman law and associated factors with transforms (l.d) is now obvious. In section 5 we shall prove the following theorem
Theorem
(1.1) :
Let
be a
with a constant drift vector X
(l.j)
where 80
t
=R
t
(8(t),t> 0)
= v!lvl,
v
BN
.
1n
d R •
0, and let
d Rt
2, started at
0,
= IXtI.-Then -
8(foo ds!R 2) s
t
is a
EM
in the unit sphere
Sd-I
and independent of the BES«d-2)!Z.lvl)
d
of R , starting (R ,t > 0). t -
process
The reader is warned that it is critically important in the above theorem that
X starts at
O.
This result is to be compared with the classical skew-product for Rd-valued
EM. (Ito - Mc Kean [Z8) , § 7.15), which
ii.s)
when v
= o.
Amongst other things, the skew-product representationVexplains the result of Reuter, mentioned in the discussion of Kendall's paper
[3OJ ,
that
for Blf with drift. with T as in (l.g) , the hitting angle 8(TX> is x independent of the hitting time T. Indeed, it is plain from (l.j) that x 8 (T x) is independent of the whole radial motion prior to T X I which improves the result of Wendel
of
Tx'
(48).
and the same holds true for
Lx
instead
289
Inspection of (l.h) and (l.j) reveals that we have a new representation of the result of Hartman and Watson of
0(T x )
on the circle
that for
d
=Z
the distribution
(which is von Mises with
ox)
is a
mixture of wrapped normal distributions, as well as the corresponding results for
d > 2. By the skew-prouJct representation of complex
ffi1
with no drift, another such representation of the von Mises distribution on the circle,with parameter
ay/t, is provided by (l.e).
We note here the remarkable fact that in all these representations arising naturally from
the mixing law is the
N1
same. Remarkable, because,
as will be shown in SectionlO to settle a question raised by Hartman and Watson themselves, the mixing law is not unique. Also in Section 5, we show how the invariance of the family undertime inversion, discovered by Watanabe enables one to give a simple representation of the
Bessel bridges in terms of this family. Consequently, the
second Hartman law and the factors whose transforms appear in (1.0) and
(l.d) may be reinterpreted in the context of Bessel bridges, but the actual translations are left to the reader.
2 appearing in the formulae s above transform very simply in this representation, which helreexplain the It also turns out that the integrals of
dependence of the formula (l.e) on
ay/t
ds/R
alone, and the resulting
ubiquity of the first Hartman law.
In section 6 we leave Bessel processes for a while, to develop a simple general formula for the density of the (infinitely divisible) law of the last time
y that a transient diffusion on the y, but we return to apply this result to Bessel processes
line hits a point
L
(R,t > 0 ; pV) t x x, we recover the result of Getoor [13J that
in Section 7. In particular, if at
(l.k)
with (1. Z)
is a
BES(V,O) started
pV(L e.dt) = (l/Z)V[f(V)tV+1r l exp(- yZ/Zt)dt, a
V
y
Eo exp(-
o.Z
2:
Ly) - Zf(v)
-)
(o.y/2)
\}
290 There are equally explicit formulae for Let
KY be the probability on V
of (l.k). (Note that be denoted
BES(V,O). defined by the right hand side
(0,00)
is just as scale
2y2
3ud that
will
in Section 9), The 1,::>.,,18 seem to have been first V encountered by Hammersley [I!l, who showed thar
KY is the
(l.m) where
K
v
W distribution of fT o XP-2 dt for x x
0
Wxgoverns the real valued
is the hitting time of a for this on the line starting at has the Student
x, and
(X ,t > 0)
ffi1.
These laws appeared again in Ismail
t
0, independent of
-
R
t-distribution with
starting at
(E(t),t
is implied by the infinite divisibility of
is a
0)
IV' K
T
a
BM
.lith distribution then v 2v 2 degrees of freedom, and
that consequently the infinite divisibility of the Student this infinite divisibility of
(y!2V)2V,pcl!V
BM
and Kelker [25J, Where it was pointed out that if
12 B(R)
=
t
Grosswald
t-distribution
[IS]
es t ab l i shed
by an analytic argument, subsequently
simplified by Ismail [23J, but these authors seem to have been unaware of Hammersley's result (I,m), from which the infinite divisibility of
KY
v
T for a < x, qimilarly, the a infinite divisibility of is plain in (l.k) by a decomposition at L b for b < y. The connection between the two representations of in (l.k) is obvious by decomposition of the integral at
and (l.m) is provided by representation of Bessel processes used in Getoor - Sharpe
[i 5J
and a time reversal. Indeed, Getoor and Sharpe
rediscovered (l.m) by remarking that if the process
(2P-t xP!2, t
an d
y
Sharpe
= 2P- 1
IJ
P!2. wh'l e t h x e '
-v
started at
reversa 1 resu 1 t
0
f
shows that this latter process with index
from the time
I
Y, where v 'II'
y
0
< T ; W) 0 x
-
= P-I
"49J or
-v when reversed
A(T that it hits zero is a BES(V) started at o) killed at the time L = A(T) that it last hits y (see
Remark (4.2) (ii) below).
-
t 0-2 At = 0 Xs ds, the
in (I,m) is time changed by the additive functional result is a Bessel process with index
°< t
°
an4
(3.3) and
291
In section 8 we apply Watanabe's time inversion theorem to obtain the distributions of T
for a
= inf{t
y
BES(Jl,o)
Hannnersley
process
[:9J
R. In particular, the fact. noted by both
and Ismail and Kelker
[25], that
of the inverse of a ganma variable with index implies that
T y
for a
BES(j.J,O)
V
is the distribution 2 and scale y /2.
has this gamma distribution. We also
encounter an infinitely divisible distribution discovered in a different context by Feller
[9].
Section 9 is devoted to the probabilistic interpretation of certain ratios of Bessel functions studied
by
[24J
and
and Kelker
[26J
we give new proofs and extensions of many of their results.
a,v > 0
In particuldr, we consider the functions of (1. n)
C.)
1
Iv ( Iii)
.;a
IV-I (! O. For =t 5-
and a
BES(v)
from
[sO.
started at
denote the law on
of
v
by
is the following Proposition. which is a slight
refinement of Lemme (4.5) of
Proposition (2.1)
Let
[59.
a > 0,
V
> 0
and let
T be an
time such that EO TV / 2 < co. a
(2.a)
Then pV (T < co) ..
(2.b)
(2.0)
(n, F=)
pV. We recall now a number of basic results x The key result for comparison of BES(v) processes with
different indices
x
F .. cr{R • s > O}. .. s-
a
on
F
=T+) dpo a
I. and
Rr Yexp(-
.. ( ;-
1
-
2
V
2
J:
2 dslR ).
s
(
t)
stopping
294
Remarks (2.2) I) The probabilities
for
0
Watanabe
[42J
and
o
pV are mutually singular on 0
v, because
with
shows that the
(3.3) (i) of Shiga and and BES(V)
processes escape from
zero at different rates. 2) Similarly, for
0
are mutually singular on
F(
with
p
+v
the laws
= t,oo
) : cr(Ru ,u > t). This follows from the -
For
T bounded this is just a restatement
previous remark, by time inversion (see theorem (5.5)
FToof of the proposition: of Lemme (4.5) in po martingale a
defined by
t
LV., (Rt/a)V t and observe that
v
(L t "T)
below).
T satisfying (2.a), consider the
To pass to
LV
pll and p V a a
exp(- -I V2 2
r 0
2
ds/R ), s
is uniformly integrable. Indeed
v
v
sup L < sup (R la) , t 0
(2.f) on
where Conditioning on pA and a (2.g)
a
Rr
now shows that the distributions on l{+ of
Rr
under
are mutually absolutely continuous with density A P
dr)
P a (RTE:
dr)
:
=
(
)
r
EaLexp (
I
2
2 v cT)IRr = r
]
•
On the other hand, from Molchanov [37J ('H.e also Kent [3IJ) we know that the BES(v) process has transition density (2.h)
296
By comparison of the two formulae
one immediately obtains the result
of tiheoreme (4.?J in [51J. namely for r;
t z:«)
I
1
(
0
] =I At(ar)
2 C ) Rt=r t
/ I Wt' (ar)
By a similar application of Proposition (2.1), the formula of Kent
[3U
for the Laplace transform of the hitting time
of
BES(v)
develc?s into the following joint transform of
which was given as theol'eme
Proposition
(2.3) :
(2.j)
a
in [51J :
(4.10)
S.b > O.
For
expr-
1.
L: 2
\)2
Tb and C(Tb) .
C(Tb ) -
1
2"
c:e = K
2
a
for
jbJ Ji (bS) -rab < a, ce.. I
(as)
TJ
A
a < b. In
particular
(2.k) where
E:
+ 1
if
b < a,
E:
=-
1
for
a < b.
From the formula (2.j) it is eesy to see that the Hartman laws with transforms(l.c) and (l.d) can be described as the distributions of C(Tb) for a BES(O) procesr started at a and conditioned to hit another level b before an independent exponentially distributed random time. We return to this point in Section 4 after first considering this kind of conditioning operation in a slightly more general setting.
297 However, we feel that, since our Cameron-l1artin-Girsanov type
(2.1) is one of the keys to our results, we ought to give another simple application of it, before passing to (perhaps) less standard manipulations later (see section 4 for Indeed, we now show how
formulae
(&7J)
may be deduced from
Proposition (2.1). Let
EM .
(Bt't .2::0) be a
r = Ix I, e = x/r, R t problem of calculating
Ed • where
IB t I,
8
t
d
2, starting at
x;
O. Put
and consider for example the
B/R t •
(2.l) where
T TaA T is a b is the first time R hits a or b, and spherical harmonic of degree n. By the skew-product decomposition of
(see Ito - Mc Kean [28J, p. 270), one has, for (2.m)
E(S.Q,(8 ) n t
C
t
IRu ,u
> 0) = s.Q,(e) expC
-
n
]lr ErLexp(21
(2.n)
A=
t > 0 :
i- n(n+d-2)C tJ\
is the clock defined in (2.d).
On the other hand, from
where
\:
X
(]l
2
(2.1) we have that for V
2 C T
1
s 2 T) ;
R.r = a]
2 1/ + v) 2 , and we hope the reader will forgive
us for using
the same notation for the co-ordinate process R as for R = IBI. Thus, ]l = (d-2)/2 and v 2 = n 2 + n(d-2) in (2.m), it emerges that the
taking
expectation (2.Z) is identical to (2.0)
where
A
(4-2)/2 + n.
298 Finally, this last
expectation can be calculated from the kno\Yn Laplace A
transforms of
T and T under P by a routine application of the strong a r b Harkov property (see Ito - Hc Kean [28J p , 30), and it is found to be
(2.p)
la
'r-)
A
IA(bs) KA(rs) - IA(rs) KA(bs) I (bs) K (as) A A
Substituting (2.p) for the
pA r
I (as) K (b s ) . A A
expectation in
(21.0)
now yields the
formula (9) of Wendel [47J, and we leave it to the reader to check that the other formulae of Wer.del
be obtained in exactly the same way.
299
3.
DIFFUSIONS. In this aection let {x ,0 < t < t -
r; < -
00
;
P ,xE:(A,B)} x
be a regular diffusion on a sub-interval
(A,B)
of
[-oo,ooJ. To avoid
unnecessary complications, we assume that
(3.a)
= inf{t
r;
>
°:X
t-
= A or
B},
so the process is killed when it reaches either boundary. Given
a > 0,
we wish to record some basic results concerning the diffusions xe (A.B)} and
xE(A,B)}
diffusion at a constant rate hit
B in the t
obtained by first killing the original
a. then conditioning this killed process to
case, and to hit
A in the -I-
case. Since the original
process may never hit these boundaries. as for example in the application to Bessel processes which we have in mind, this conditioning is to be understood in the sense of Doob
r 6]
and William8
[3.9J. Following Hilliad' s
description of this operation with no killing (i.e. a
[!.9],
of
= 0)
in section 2
we take
pat to be defined by the requi rement that for each x x < b < B. the process X run up to the time T has the same law under b pat as it does under P conditional on (T < Va)' where Va is an x x b exponentially distributed killing time with rate
Putting
m
cr(Xs,O dpat
(3.b)
_It_
dPx
=e
a
(x,y)
independent of
t), this is just to say that for
S
-or b
I
¢", (x,b)
'"
where (3.e)
a
= Ex
on
X.
x < b < B
b
e-aTy.
a. 0, e-aTy should be interpreted as the indicator of the event (T y < (0). Note that ¢a(x,y) > 0 for all a 0, x,y E.(A,B) by the
For
assumption that
X is regular.
300
Before going further, we recall some well known facts concerning which may be found for example in either Take a point
xot!; I
Ito
¢a(X,y),
and tolc Kean [28J or Breiman
[3 J.
and define
ts.a:
y < x -
• J/¢ (x ,y),
a
0
0
y > x o'
8i nce the identi ty (3.e)
is valid whenever
y
x
z, one gets :
(3·f)
which shows that the choice of reference point
affects ¢at only by o a constant factor. Similarly there is a function ¢a+ which gives the
analog of (3.f) for
x > y. These functions ¢at
X
and ¢a+
may be determined
as solutions subject to appropriate boundary conditicns of the equation (3.g) where
(G-a) ¢ = 0,
G
is the generator of the diffusion.
Proposition
(3.1) : Let
T be an
time,
drat x
on
x
ProOf: For
f\(T < 1:;)
T Tb, this follows from (J.b) after conditioning on using (3·f) and the strong Harkov property. For general T consider T AT let b,
b
tend to
Band use (J.a). 0
301
It follows easily from the above proposition that the prObabilities {p
define a new diffusion process which is transient with
x
is.n: except if
a
=0
and the original diffusion
{Px}
is recurrent, when
pot • p • x x pat x
Clearly, the probability all
x,
< 00)
is either 1 for all
x or 0
for
Since
(3.i)
Eat x
= lim Eat e-
STb
b+B x
at motion hits B in finite time a.s. iff the limit lim $ (b) $ (b) is strictly positive for some (or equivalently b+B at a+S,t' nll) S > o. the
I
Let
Pt(x,dy)
Then from
be the transition function of the original diffusion. (3.1), it is plain that the
at
diffusion has
transition function
(3.j) A formal calculation based on (3.j) shows that the gene ra toc d:xt must be
Gat
(3.k)
=
and in particular if d
2
d
G = a(x) ---2 + b(x) --, dx dx a further calculation using (3.g) reveals that at d (3.1.) G = G + 2 a(x) ¢at(x) dx' where
=
¢at· As we shall not make any use of these formulae foz
generators in what follows, we shall not attempt a careful justification, but rather refer the reader to Kunita
[33J
and Meyer
are deftly handled in a much more general context.
[3{] ,
where such matters
302
Of course, after some obvious substitutions such as everything
above applies equally well to the
conditioning
X killed at rate
a
at
for ¢at'
process obtained by
to hit the lower boundary point
A
instead of the upper boundary point B. As the reader can easily verify, we have
Proposition (3.2) : If either the the result is the
(a + B)t
In particular, taking
or
at
process is conditioned.
St,
process.
B = 0,
we see that the
at and
are dual in the sense of section 2.5 of Williams
processes
As a consequence,
either process can be presented as a time reversal of the other. To be precise, for
y( (A,B). let
(3.m)
L
Y
sup{t
y}.
Then we have
Taeorem (3.3) Fix
>
o.
theorem (2.5))
(William!:>
Suppose that the
at
process hits
A in finite time with
probability one. Then A is an entrance point for the y
(A, B)
at
process, and for. each
the proces ses {X(s-t). 0 < t
{X(t),O < t < Ly
are identical in law.
Remark : Williams
states this theorem in the case
a
= 0,
starting from
a process satisfying hypotheses which make it identical to its own Ot process, and with the roles of
A and B reversed.
303
However the apparent extension above to a general superficial one. by
e
(3.2) with
e
a > 0
is only a
O. Williams proved his
theorem by first establishing a special case and then arguing that the result could be
to the general case by the method of time
substitution. The result can also be deduced from the time reversal theorem of Nagasawa [33J. via the work of Sharpe
[41J.
The connection wi th Sharpe's work is
easily made after noting that
(3.n) serveo as a scale function for the
at
process. with
s(B-)
=0
and
s(A+) = co.
The reader should be well prepared by now for the conditioned Bessel processes of the next section. But lest our change of hitting rate from
a to
02 , and our use of
the tenn"drift" for 0 in that section seem mysterious. we recommend the following trivial exercise
Exercise (3.4) : Show that. for Brownian motion 15 > 0, i)
e
ii)
tbe} 02t
iii)
changing
iv)
for +
v) (Hint
t to
EM with drift + 0
+ above changes
BH with drift
Y. the
-!-
+ to - ;
02t
the recipe iii) applies to iv) too. use (3.2).
on the line with zero drift.
+ox
process is
/y2 + 62
(BM)
process is
BM with drift
304 To conclude this section, we record the following result, which will not be required
the end of section 4.
Given a random time
= O{Ft(t
L, define
< L), FtC [t,t
OJ,
and let L be as in (3.m). y
Theorem (0.5) :
that the regular diffusion
meaning that for dpat x dp x
is.«)
(3.0)
= c(x,y,a)e
-aL y
=
I. Then ----
on
=
is transient,
x
()(Ly > 0),
Y
c(x,y,a)
Proof: For
P (L < 00) x y
x,yE (A, B) ,
-
{p }
where
-aL 0) / E e x y x
-aL Y (L
y
> 0)
t = 0 and F Q above and t" by the regularity and transience of {p}.
where (J.q) is obtained by taking c(x,y ,ex) E' (0,00)
= 0)
pat ' ht x a.s., an d on t h e r1g so it follows that (J.n) holds with
btB
(J.q)
) pabt(L y
-aL b y)
-a(Tb-L b) y I Lyb > 0)
b t B. On the left we have
L btL ,
Y
Px(e
y
e
x
305
Finally, to turn (3.q) into (3.0) use (3.f) and the obvious formulae (3.'1')
x
5.. y
¢a (x,y) ¢a (y,x), (3.8)
Remark
E
x
(3.6) :
e
If
-aL
y (L
¢ (x,y) E yay > 0)
=0
e
-aL
x > y.
y
A is an entrance point for the
as will be the
case in our applications to Bessel processes, the Laplace transform of
L
appearing in (3.0) above can be computed very easily, since -aL
(3. t )
Y
-aLy
y
while by Williams's time reversal (3.5) (3.u)
whence
ts.»:
E e
-aL Y
y
I ¢ (y,a)] I [lim ¢ (a,y)l.
:[lim ¢ (y,a)
a+A
a
a+A a
0
.J
As will become clearer in Section 9, it is interesting to ask what can be said along the lines of Theorem (3.5)
the basic diffusion
is recurrent. To focus on the most important case, fix the probabilities pat on _ for a > o. y
Note that
y
y
is
for
a > 0,
o
for a
a,S> 0, from (3.8) and (3.5) we have
is,»
c(a,S) e
-(8-a)L
y
on
-' Y
where at c(a,S) : c (y,y,S) and, as a consequence, for
I
a,S,Y > 0
c(a,Y) : c(a,S) c(S,Y).
IE
at y
e
-(S-a)L
y,
x
y > A and consider
y
pat(D < L < 00)
{p}
O. Still, for
306
Our analogue of Theorem (3.5) for this recurrent case is
Theorem (3.6) : Suppose
{p } is recurrent. x
y > A. there is a strictly positive function
For each
cr-finite measure
My
such that H (L
Y Y
00)
a
a
0
y
_'
a
f (a) y
and a
each defined uniquely up to constant multiples.
and for every
a > 0
dpat -at y - L .. f (a) e y
O. One can take
Then for
f
y
_. for every
(a) .. t/c(a,Y). a> O.
y
(3.x/7.)
M (A) .. c(a,y) Eat(e Y Y
(3.xb)
aL y
A)
.. lim c(a.y) pat(A). y
a4()
Proof: The fact that (3.xa) defines a measure which does not depend on a is immediate from (3.w). The rest of the assertions follow at once, using -aL M (A) .. lim M (A e Y) for (3.xb). y
o-o y
Remark (3.7) : It follows from Proposition (3.1) that for an arbitrary diffusion
(P x),
and for any
a
the Law
BR(x,y. t)
of the bridge obtained
given X .. x and X a y 0 t is the same for either the at. ai. or original process. As a consequence,
4S
the conditional distribution of
0,
(X.O < S
-
S
< t)
-
at process at time be described as follows : under M , y
using the last exit decomposition of the a-finite measure Hy
y' the L has y
L
a-finite distribution
M (L Y
Y
dt)-f (a)-t eat pat (L Y
Y
e
Y
dt)
for any a > 0, and, conditional on Ly a t, the process (Xs,O 5 t) is a BR(y.y.t). For an even simpler description of M in terms of local time y at y, see Remark (3.9) below.
307
EXample (3.8) : For
0, let ll) 0 ; W
(Xt' t
be a Brownian motion with drift
started at zero, and let
sup{t : X = a}. By the method of time inversion used in Section 5, t one finds easily that L has a gamma distribution, with
L
(1,
1 02
';(e-
"7" f>
L) = (I +
and
2
1
L)- .,.
i
2 t/2
e
since
+
is the
from
(3.6) that the M(A)
= 11
-I
11 E [exp(z L)
_
; AJ
M governs
(3.9) :
=I
1
dt,
M(LEdt) = (2trt)
(Xs'O
2
s < t)
as a
bridge.
that in the last example we have My(LyEdt)=pt(y,y)dt,
(3.y)
where
is the transition density of the diffusion. In fact this
Pt(x,y)
holds quite generally, as a result of the following description
formula
M, which the reader can easily verify using (3.1). UnderM y
local time
100
at the point
Lebesgue measure on (X ,0 < t < L ) t Y
Tu
defined by
the nice constant is obtained by taking y
in (3.xa). Thus
of
(see (3.4), one finds
W
a-finite measure on
does not depend on
and given L = t,
O
process obtained from
= infit
y
(0,00). and conditional on
: 100 = u}. Thus, the M
y
M (0 < L < t) -
y-
a
0
100 = u
(X ,0 < t < T t
-
-
u
distribution of
the process
; P ), where y
L
y
is the potential
0), up to a constant c > 0, whence,
(Tu'U
cE [I(T Y
the total
has distribution which is a multiple of
has the same law as
measure of the subordinator,
Y
y
< t)dt
u-
= cEy
[1(1 0
t
>U) dU
= cEy
308
If
My
and
(2 t,t
0)
are appropriately normalised this leads to (3.y)
for any diffusion with sufficiently regular transitton function -see e.g. Getoor of
My
D4J,
and (6.d) below. We note that this last description
makes sense with
with arbitrary state
y
a recurrent point for a strong Markov process
309
4. CONDITIONED BESSEL PROCES5ES.
v,o
For from the
BES(v)
processes obtained
diffusion of Section 2. From
(2.3), it is
0 > 0, x > 0, one can take
plain that for (4.at)
+02t and +
> 0, consider now the
2 (x)
¢IV
i-
0 t
(4.a1)
2 (x) .. x
-V
K (ox), V
and V
(4.bt)
¢lOt (x) .. 1,
(4.M)
¢1
where
V
o0}(x)
.. x
-2v
x > 0. The results of the last section reveal that for
BES(v)
t
diffusion conditioned
02 t
to be referred to as course
BES(V,O)
is just
0 >
is given for
°
v 0
(4.ot)
the
with infinite
[0,00)
BES(v,o). Of
the transition density
of
BES(V,O)
by
Pt' (x,y)
which shows that our
°
BES(v).
and (3.j)
From (2. h) ,
a diffusion on
BES(v,o)t, or simply
0 >
yt
-1
BES (v, 0)
I (ox)
v
-]
2
2
2 2
I (oy) I (xy/t) exp -(x +y +0 t )!2t,
v
V
is a process introduced by Hatanabe [46J, and
called by him a Bessel diffusion £rocess in the wide sense with index a .. 2v + 2 is the "dimension", and c '" 0 2;2.
(a,c) ,
where
Remarks (4. 1)
(i)
Watanabe allows his
which corresponds to be extended to
V
> -
a
to be any strictly positive nunmer,
v> - I. The above definition of
if the boundary point
0
of
BES(V,o) BES(v)
can also
is taken to
be reflecting, which completes the correspondence with Watanabe, but the reader is warned that because the assumption (J.a) is no longer satisfied, the results of section 3 must be reinterpreted with some care to cover this case.
310
(ii)
A further extension of the definition of
nothing new. Because
BES(V), it follows from (4.2) (ii) below that
and
to
V
0
and
8 >0
this process rear-hes
Therefore the
process started at
Williams theorem (3.3) and run to time
0
x
Lx. For this reason, results for
playa dominant role.
v
=6
= 0, for all
in finite time and dies
as the time reversal of a
be reexpressed in terms of
call it, there.
can be described via EES(v,6)
started at can readily
BES(V,O). and it is this process which will
0
311
Rerrm>k (4.2) :
For the sake of completeness, we record the following facts about BES(v,8H. (i)
The
version of formula (4.0) has
K instead of
I
in the
first two Bessel functions only. (ii) In the
case of (4.0), (x/y)v
for
and (S.Z),
these two factors. By inspection of BES(-v)
should be
is just
killed when it hits zero, a fact which is implicit in Sharpe
(iii)The
version of (4.d) has
K instead of
I
everywhere, and
a factor of - 1 in (4.dS). Thus from (4.k) below, the extra drift term in this case increases from (iv)
is
to - 8 as
BM \vith drift
x increases from
0 to
- 8 killed when it hits zero,
a fact which is intimately related to remarkable properties of
BES(t,8)
described in [4Q1 and [49]. For
8 > 0, v,x
on the space
0, let
C(R+,R+). and let
the reader that the "8"
v 8
Px'
x
be the law of BES(v,8) correspond to
refers to killing at rate
we should now declare that
BES(v,8)+
t
started at x Qe remind
8 2• Strictly speaking
is absorbed rather than killed on
reaching 0, to keep the trajectory in COR+JR+), but this won't ever be impor.tant. By a straightforward
application of Proposition (S.l), we obtain
the following extension of Proposition (2.3) to Bessel processes with drift :
312
Theorem (4.3)
x.a,r > o. v
Let
_
(1.e+)
where
exp [- '2 a
Iv(ar)
2
t
!£e('YX)
B TrJ = I ( 8x) \i£e(yr) v + ( 2 ) 1/ 2, and .;;f?= I. if x < r
C(Tr) -
e = (i
y =
if
O.
2
!
K,
x > r,
The corresponding formula (4.e+) has
K substituted for
in the first
I
ratio of Bessel functions only. By Williams time reversal. for
r < x, the
expectation on the left side of (4.ei) is identical to exp r, L:
RemaPk
+i
C(L ,L ) -
r
x
t
2
6
(L
x - Lr
)J
(4.4) :
We note from (4.et) the formula
va
(4·f)
P'
x
(T
r
< co)
= Hv.s ' (x)
».s
/ H • (r), r < x ,
vlhere
(4.g) a result which is also obvious from (3.n) and (4.a).
a Thus - HV'(x)
serves as a scale function for
J:
expression (4.h)
y
BES(v.a)t. The alternative
du / U(I (Ou» 2, v
which is the equivalent of forii1Ula (2.5) in Watanabe [!16j. is a simple consequence of the classical fOITaula for the Wronskian : Iv(z)] = z-I
From well known asycptotics of Bessel functions which are displayed in Section lit one can now obtain the asymptotics of (4.et) when and those of
when
r -+- 0
or
x»
r -+- 00
or
x-+- 0,
0:>.
In particular, one obtains the following formulae. the first
of which
imply the interpretations (1. h) and (l.i.) of the Hartlz:!an laws. and the third of which is equivalent to a result of Kent by virtue of
CopoZZary (5.6) below.
([31J.
theorem (4.1)),
313
Co!'oUOX'!J (4.5) :
EV, 0
(4.i)
exp [;
0
EV'o
t a?
exPti- a
0
2
C(Ty , 00)1 .. 1 C(L
y
0:
/ IV (oy) ,
e(OY)
K (oy) / Ke(OY),
V
e .. (V2 + a 2 ) 1/ 2
where
t t exp -
J
I
s\
(6Y)
(4.j)
V' o Vl2 1 S2 T " I v (yy) (1 + E0 exp - 2" y 2 ' 0 V
(4.jll)
E'
V 0 o
2 2] 1. S L .. - - - (I + lL)V/2 ' 2 Y Kv(OY) 02
y .. (0 2 + V2)1/ 2 •
Proof:
Proceed thus from Theo!'em (4.3).
(i)
In (et) put
(ill!) In
(j)
put
In (et) put
x .. y. 6 .. 0 r .. Y. 8 .. 0
.::
{4.k}
I
o.
and K
-\l:ll
and let
x
00,
-+ co,
x ... O. r ... O.
define
(x)" I (x) / I (x) ; K V
We note that on putting
lJ:V
and let
x .. Y, a .. 0
(x)" K" (x) / Kv (x), x > O. >-'
the formulae (4.i) and (4.i_) above make obvious
the result of Hartman and Watson I
r .....
and let
r .. y. a = 0
(j:r:) In {e+;w;} put
For
and let
• IToposition 7.1, th3t for
are continuous distribution functions on
It now emerges that for fixed 0 > 0 and x > 0, the o for pll• 0 are mutually absolutely continuous. x To be precise, we have
both
(0,00). laws
314
Theorem (4.6)
Let
8,x > 0,
O.
(4."l)
on
and for every
(!t+)
stoP?ing time
!oo'
T,
(4.m) Note
on On
(T
= (0),
a
1
by convention.
Proof: It is enough to concider the case For bounded stopping times
T
and prove (4.m).
the ref.ult follows at once from
and (3.1). To ey.tend to unbounded expression above. Since
> v
(Mt,t
T 0)
let
denote the right hand
is an
under
it only remains to show that this martingale is uniformly integrable, or, what is the same, that its almost sure limit as expectation equal to
t
4
00
has
1. But this is immediate from (4.i)
mentioned above that
increases to
as
z
4
pV.8 x
and the fact
co,
Com nary (4. 7)
Let
Z > 0
be an
random variable. For fixed
x,8 > O.
the function V
v , 8 Z, v > 0
E
4
x
is right continuous, and continuous except possibly for a jump at VD inf{v : Ev,8 Z < co}. X
from
co
315
Proof: Use (4.6), the continuity of
I (y). and the monotone and u
dominated convergence theorems. Note
0 DOor x· O.
The above result is clearly false if either
T. 00, condition on f L
Proof: Starting from (4.6) for exit decomposition at time
Corollary (4.9) : For
x
together with (4.i-).
> 0,
()( D
L y
)ll-V
_. and use the last
"I
r +(u 2-v 2)C L 1
exp[
Proof: This results from (4.8) on letting 0 passage to the limit being justified by (3.5).
Y
0, using (13.£),
the
316
5. n:E RADIAL AND ANGULAR PARTS OF BROWNII\N MOTION WITH DRIFT. This section offers two different approaches to
Rd
BN with drift in
and its decouposition into radial and angular parts, using firstly the
Cameron - Hartin formula, and secondly tiee inversion, to transform to the more familiar case with no drift.
D. Williams seems to have initiated the use of the
- Martin
(Clf) Eorraul.a to ca l.cul a r a distributions associated with the radial and angular parts of EM with drift (cf : the end of Kent's paper [31]). The method
Ls also
(D'O, [32J,
used more or less explicitly in a number of recent papers
[48})
but in none of these papers is the argument
developed to its fullest extent. Fix an integer
d
2
I, and for
0
0, let
the canonical realisation of Brownian motion in R with a constant drift
=
....
0 of magnitude
....
a =101
the unit sphere in Rd. Thus
Bt(W) = w(t), =a(Bs'O 2 s 2 tl. {Bt,t d R starting at the origin, and pO is the
....
(Bt + to,t time
0). According to the
d
a;
p8}
be
started at the origin
in the direction
n=
C(R+ ,Rd),
0 ; po}
po
is standard
PM in
distribution of
CM formula, for any
stopping
T,
(5.a) where or
(
,
)
is the inner product in
Rd . (See e.g. McKean [29J p.97
Freedman [llJ,§l.ll). The applications of the 01 formula below hinge largely on the product
form of the Radon - Nikodym derivative, which can be exploited by virtue of the following general (and trivial)
317
Lemma (5.1) : Let
(n,V,
P
Q be probabilities on a measurable space
and
with
where
G
>
(i)
dP
°
= GH
is
For
Z
= =
%
and:
0, Q a.s.,
/
and
P-independent, they are also
Q-independent,
P(G) P(H) =1,
G/PG on
=
0, let
k
at
H/PH
= U(d8)
vM(k,d6)
(S.b) U
= vM(O)
on
1:e.
be the von Mises distribution on
vM(k)
concentration parameter
h were
..
is 0
H
Sd-l
centered
k. That is,
Cd(k)
-I
....
exp k(6,u),
, th ' f orm pro b a b 1i Li1ty on 1S e un1
Sd-I , and
Cd(k)
is the
normalising constant (S. (1)
Cd(k)
(5.a2) where
V
=
= fU(d6)
exp
=
(k/2)-V Iv(k),
r(V+I)
(d-2)/2, and the formula (5.02) will be later derived in (5.4) (iv).
Starting from the easy case and the CM formula which is implici t in
[4cD.
Proposition (S.2) : Let 1
T be an
0T given
'R T
i) ( ' YV
.£!!
0 = 0, using part (i) of the Lemma above
easily obtains the following proposition.
)
t
R =
Ist I. jRt = a(R s ,0 < s < --
stopping time. Then. the
(T < (0)
is
t), 6
t
=
&t/Rt • and
. P6 cond1tional law of
318
Now for pO
T r
= r}.
inf{t : R t
it is obvious by symmetry that under
the uniformly distributed angle
radial process for all
(R.O < t < T) t
-
-
8
is independent of the stopped
Tr
:R T .
generating
r
Moreover,
r
/> (T r
< co) .. 1
0, r > O.
0
Thus part (ii) of the Lemma and the CM formula (S.a) imply Pr>oposition (5.3) : Let
where
{Bt,t
origin,
Rt
(i)
p6"}
0 ;
=
IBtl.
is a
R1
8
The hitting angle
The
(ii)
p8
with drift
inf{t : R .. r}, t
in
starting at the
and the radial process up to time
T
distribution of
8
T
vM(or}.
is
(R ,0 < t < T ) t
-
r
under
po
is identical
conditional on
Uo is an exponentielly distributed time with rate Po
Tr •
r
to the distribution of the same process under
which is
R
p8
(iii) The distribution of
(Tr < Ud),
6"
T r
8 t .. Bt/R t•
t < T ) , are r
(Rt,O
r > 0, and let
+ 02 •
independent of this process.
Remarks (5.4) : The first four remarks refer to the correspondingly numbered assertions above.
Stern
for
(i)
This extends independence results to be found in Kent
(43).
and Wendel
(ii)
This may be found in Kent
GS].
But. see (v) below for a further extension.
d - 2. The joint distribution of
obtained by Reuter (see
5OJ).
I'IJ, and in oT and Tr r
Gordan and Hudson
[171
in this case was first
319
(iii) An immediate consequence of this is the result of (Rt.O
t
O,
(vii) By a further application of (S.l) (i) to (S.a), for
cs.eu
the pt conditional law of is identical to
(5.e2)
the
po
conditional law of
On the other hand, for decomposition of BM on Sdl
d
d 8M in R
e
r
t
where Uo is exponential with rate (B ,0 < t < T) under pO. t
T IB r) T
(Bt'0 2. t
t
2
Tr)le
02
T
r
=
e,
Tr < U ' o
independent of
r
2
one can start from the classical skewproduct
with no drift, use the reversibility of
as in the discussion of "spinning" in It'O McKean [28J,
section 7.17 to argue that under
po
for each
r
>
0,
321
Bt •
(5.!) where
is a
1l
r
0) is BESlvl (v) (a subscript is now being used to indicate the starting position of a process). Then, the identification is a consequence 0)
as
BES
of the following remarkable result, which will be the key to several further developments. Theorem (5.5) : (Watanabe For all
V
and only if
> - I,
y,6
[.46].
theorem (2.1».
0, a process
(t U(l/t),t > 0) is a
(U(t),t> 0)
BES
i!-!
BES
y(v,6)
if
6(V,y).
Watanabe's time inversion (5.5) and the Pythagorean property of the BES(V)
family, discovered by Shiga and Watanabe
Corollary (5.6) : Let an independent (5.j) BES
o
(Xt,t
0)
where
be a
imply the following
BESo(A,a), and let
> - I,
a,e
(Yt,t
0)
O. Then the process
be
323
0';: 0, \I > - 1/2, if
In particular, for (Be t ,;: 0)
is a
BMo
process in (5.jJ is a
-
and
It]
X t
• IB t
+ otl
whete
is a BES (\I -
(Yet t ,;: 0)
--
0
0 >0
and
then the
2
BES (\1,0).
--
0
This last presentation of
BESo(\I,O)
underlies the work of Kent
Proof: Let At = t X(l/t),
Bt - t Y(l/t). By Watanabe's inversion, A and B are independent BES (A) and 2) processes. Now, by the Pythagorean property of [42J, + B 112 is a BESy(\I) with starting place y = (a 2 + s2) 1/ 2 and index \I = A + + (which corresponds to adding the dimensions). Inverting once more yields the desired conclusion. We
how
Watanabe's time inversion can be used to obtain a very simple description of Bessel bridges. Somewhat more generally, consider a family of diffusions d indexed by a parameter YES in such a way that on a subset S of It if
pX
governs the co-ordinate process
starting at
0, then the laws
Inversion Hypothesis (sX(I/ s) , s
EXample 1:
>
(S.
7). For
(Xt,t,;: 0)
{PX,y,OES}
is pO t., h ! y' we ave d S _It and governs
pX
y-diffusion
satisfy the
y,o ,,-S, the
0)
as the
distribution of
examples in mind: E1
with drift vector y
started at
Then (5.7) is a variant of the familar time inversion property of
Example 2:
S =It+
and
pX
governs
BESo(V,y), where
V>-
O.
BM. is fixed.
Then (5.7) amounts to Watanabe's time inversion. For such a family of diffusions there is an extremely simple description of the bridges obtained by conditioning the two ends of the sample path over a fixed time interval :
324
{prJ
Theorem (S.8) : Let
be a family of diffusions on
S
satisfying the
inversion hypothesis (S. 7). Let
y, tiE s. Then, the processes
t > 0,
o 0
For
y
(6.e) (ii)
x
the formula (6.e) cefines an infinitely divisible
probability dist=ibution on : (i)
x > y, the distribution defined by (e.e) on
For
If
(0,00)
is
sub-probability with total mass given by (B.a).
an infinitely divisible (ii)
(0,00).
is an entrance point for the diffusion, simple formulae
0
for the Laplace transform of the law (B.e) can be obtained from (3.v). on
r of the diffusion will coincide
(iii) In practice the 2 C (0, 00) with
=t
2 d d a(x) ---2 + b(x) dx' dx
Suppose simply that
a,bECOO(O,oo) , with
Y
hypo-ellipticity of y, a function 00 3 C on (0,00) such that
a> O. Then
dm
-)
(y),
hence = Pt(x,y) (s'a) (y),
and the formula (6.e) becomes
ie.«: )
exists, by the
p : (0,,,,,)3 3(t,x,y) .. pt(x,y), of class
In this case, one has dy =(s'a)
thb-e
327
x > 0, Y > 0,
Pl>oo[ : For
P (L
x y
> 0,
t
< c) .. E [I - u (R
x
Y t
)J.
Put Mt " s(Rt)' and recall that for each x > 0, M is a Px-local martingale with continuous paths. One can thus apply the generalised Ito (or Tanaka) formula to the process H t uy(R t) .. (s(y» AI. (Az.t >
Accordin.g to this formula. if (see Ueyer
t
-
denotes the local time of
0)
M
at
Z
[35] Chapter VI ; II), the process (u (R ) I -- AS(y) t > 0) Y t 2 s(y) t • -
is a
P -martingale starting at u (x) which is square-integrable (it even x Y belongs to BMO - see [15J tMoreme 4, p , 334). It follows that P (0 < L
x
On
Y
< t) ..
l:ll
E
s(y)
x
AS(Y) t
the other hand, Ito - Mc Kean ([28] p , 175) show that for all 3
(t.x,y) 0) be the s right continuous inverse of a local time process
A for the point
y. Then
A(L ) = A(T )
Y
where process
(J
(J
A(Ly). But by Ito's excursion theory (see e s g , Heyer [36]) the
(A(,S)'O
2
s
2
(J
Py )
has the same law as a process
where
Y is a
random time independent of
and
(J
is an exponentially distributed
Y. Since the exponential law is infinitely
divisible, the conclusion is immediate.
329
7. DISTRIBUTION OF LAST EXIT TIMES FOR In this section we record explicit formulae for the distribution of L
sup{t : R • y}
Y
when
(Rt,t
t
distribution of From (6.e
l
)
is a
0 ;
Ly
BES
y > O. and consider also the joint
x(v.8).
and the clock
C(L ) '
y
and the formulae (4.g) and (4.k) for the scale function of
BES(v.8), for all
v.x.o
O.
(l.a)
where
tS .. O.
.. y/v.
The formulae below for densities follow immediately from (?a) and the formulae (2.h) and (4.a) for the transition density of
BES(v.8). The
corresponding formulae for the Laplace transforms can either be derived
J:
from the density expressions using the well known formula
tr.»
exp(-
t (it)
.. 2y (y/x) v Iv[a.(xAY)]
or they can be obtained using (3.v). Notice that the total mass of the law of
Ly
on
(0. 00 )
can in each case be obtained by setting
the formula for the Laplace transform. Formula as
and is included only for the sake of completeness. 0,> O. x> O. v
(?a)
v
tS
P , (L Edt) .. x Y
0 :
dt Iv(xy/t) exp(- (x 2+y2+82 t2)/2t] 2t I
v(8x)
K v(8y)
ex. ..
0
in
was obtained earlier
330
a > 0,
x
= 0,
\I
0 :
(7.d) 2t (at) \I K)ay)
(7. 'j)
Case
(7.e)
0 = 0, x > 0, \I > 0
J
r (x 2 +y 2 )/2t P \I (L E.dt) '" dt(\I/t) (y/x) \I I,,(xy/t) eXPLx Y v
Finally, the case
0 '" 0, x
= 0,
\I > 0
can be obtained from either the last
x tend to 0, or the previous case by letting zero. The result is the formulae (l.k) and (l.l) of Getoor.
case by letting
'V
'V
From the Laplace transforms (7.c) and (7.e)
Corollaries (4.8) and (4.9) for change of law on
Ty ' valid for all
E\I,a{exp[_
x
tend to
and the formulae of _, we now obtain y
of FToposition (2.3) and Theorem (4.3) with
analogues of the instead of
0
\I
+1)2C(L y ) - t
0, X > 0 :
S2 L y
J ; Ly
> O}
Ie [y(x..,y») r 0
if
0 = O.
Kv 0) y y C(o) y
oy.
Hartman law with parameter
given
(0
y
d
iastead of
given
(a
Y
Ly '
> 0),
is just the first
> 0)
Obviously when
Ty
= 2v
+ 2
is an integer
one can go further and express these results in terms of the radial and d, angular parts of a inn but we leave this to the reader. From the results of the previous section, we obtain explicit formulae for both the density and the Laplace transform of identity in law of
(B.b)
1
and L , we have for all y
y
•
From
v,x,a
0,
v,x dt v,o r. ]-1 , Po (TyEdt) = t 2 PI It (x,y) LG(o,y)
t
T-
y
where here again, and in the formulae below, and
T
(7.a)
and the
> 0,
0 is the starting point
the drift. From (B.b) and (4.0), we obtain, in the
x
0 > 0, x > 0
dt Iv(xyt)
(B.c)
i- [(x 2+y2) t
+o2/t])
2t Iv(ax) Kv(OI)
v
(B • if)
where
E' o a
Xl,
a,,(x,y)
2
exp(- TAT y) = and
b"
IV(oa,,) Kv(Ob,,) Iv(ox) Kv(OY)
= b,,(x,y)
,,= x 2 + y2 +" 2 ,
2 + b2 a"
are defined by the requirements
334
or, to be more explicit
To derive
a>."
t
[(x+y)2 +
>.2flz -
t
[(x_y)2 + >.2)
b>. ..
t
[(x+y)2 + >.2J
lIz -
t
(x_y)2 +
>.2J
"». 1/z.
from (8.0) one uses the following identity, valid for
o
< a b < which results from (7.b) after substituting using (2.h). and making the change of variable u" lIt
x = a, y = b, E = 0,
The derivation of the corresponding formulae in the remaining cases is straightforward. One obtains 0, x =
° dt (y c) v eXPt
te.a)
Case
where
{j
1-
(it +
2t (j'V K ( 0, x = 0 the infinite divisibility of
and (8.'!) , which is the exhibit the remarkable fact that the Po distribution of T Y and the pv,y distribution convolution of the pV distribution of T y o o (ii)
T y
is sean using \)
The infinitely divisible law (8.e) was encountered by
Feller [9] (see also [IOJ) in the study of first passage times for a continuous time random walk : for positive integers V the law (8.e) is the distribution of the first passage time to starting at zero with jumps of + 1 at rate at rate 2y2/(x2+y2)2.
V
of a compound Poisson process 2x2/(x2+y2)2
(iii) We do not kDow if the distribution of divisible for 0, x > o.
T
y
and jumps of - 1 is infinitely
336
9. COMPLETELY MONOTONE FUNCTIONS ASSOCIATED WITH LAST EXIT TIMES OF BES(v) Our aim in this section is to explain how the complete monotonicity of certain ratios and products of Bessel functions,
of which were
studied by Ismail and Kelker [26], can be related to behaviour of Bessel diffusions prior to last exit times. Ismail and Kelker showed by purely analytic arguments that for
& > 0, there is an infinitely divisible probability distribution (0,00)
with Laplace transform in a > 0
v+
rev + 6 + 1) 26 I 6 ( /ci) ------';';"""-;;.....;:.-- = rev + I) (IcY:) 6 I V (lci)
(S.al)
and that as
[-ax
V > - I,
t , 6 on
V 6 l ' (dx),
e
0
v
v 6 + 00. 1 , 6 converges to the
1nw 1V
with Laplace transform ,-V
(9.a2)
(va)
Ir(v
where the identification of due to Kent @IJ. Here
T
y
i-ax = Joe
V,....
+ 1) 2
Iv(va)
1
V
is the hitting time of
as the y
V
(dx)
= EoV
BES (v) 0
12
exp(-aTy 2y ),
distribution of
for a
usual reflecting boundary condition at
1
(Ty/2y2)
is
process, and the
0 must be stipulated for
-1 < V < O.
Ismail and Kelker also proved that for v > 0, 6 > 0, Y > 0, there is an infinitely divisible probability distribution Kv , 6 on (0,00) with Laplace transform in a > 0 (9.bl)
and that as
_rev___-;;-_..;.v__ = [-a.x + 6) 2
6
K (/ci)
6
r (v) ( lii) K.v+6 (lei)
6 + 00, Kv,e with Laplace transform
0
e
K
V,6(d)
x ,
converges to the infinitely divisible law
KV
337
00
(9.b2)
v
f° e
K V
v exp (- aL y), 2 y/2
K (dx) = E
0
where the idenfication of KV as the pV distribution of o
Ly 12y 2
is
made by Getoor's formula (l.l). In view of the obvious Laplace transform identities corresponding to the convolution identity ,Va ,v,S * , ,
(9.a)
and its companion with
,V+S,
V
> - I, S > 0,
instead of
K
"valid for
v > 0, these results
of Ismail and Kelker are equivalent to the assertion that for each
y >
°
there exist on some probability space two processes with independent increments
(9.d)
(Tv, y
V
and
> - I)
(LV,
V
y
> 0),
each with decreasing trajectories coming down from 00 to 0, such that for . t h e appropr1ate . 1 , v , and LVI 2y 2 has each V 1n range, TV I2y 2 h asaw law
V K • Then, taking simply
have distribution
,V,s .
y
y
y
a
1/12, the increment
TV - TV+S would Y Y
Note that because the trajectories have n finite limit at infinity but O V not at their start, this increment would be independent of T + but not y
of
TV, in contralt to the usual case of a process with independent y
increments starting at zero. It would be interesting to find a presentation of
BES (v) for varying o
even for all
V
in which such processes were embedded, (perhaps
y with independent increments in
(y,v)), as the results of
Ismail and Kelker would then follow immediately from those of Getoor and Kent. We do not know of any such representation, but in the course of our investigations,we shall provide probabilistic proofs of the existence of
,v,S
for all
V > - I, S > 0, and of
Kv,e for all v > 1, S > O. The
gap in our argument for the second case if
0 < V< 1
is curious, but
stems from the fact, obvious on differentiating (9.a2) and (9.b2) at a
a
0, that
(9.el)
338
while (9.62)
V Eo Ly
c
y2/2(V-I)
- I, so the Laplace transform of tV (9./)
exp[-
J:
(I-e-
ooc)
,
ex > O.
AV(dxl
is decreasing,
V
in (9.a2) is
tt is clear that the existence of laws
(B.g)
t
t
V,e
satisfying (9.al) is
> - I,
V
(where'1:iecreasing' means "decreasing when evaluated on any Borel set") since t v,6 then appears as the infinitely divisible law with Levy measure v+6
V
A - A
..
V
• But we can determ1ne A
differentiating with respect to a
by taking the negative logarithm and
in (9./) and (9.a2). After using the
recurrence formula (.13.4) for the derivative of
(9.h)
[
o
Iv' the result is
I (k) x e-nx AV (dx) .. --,,\>+-::..1- - _ , a > 0
21ci Iv(k)
which proves that (9.al) holds for
V
> -
I,
e ..
I, with the measure
tV,I(dx) • 4(V+I)x AV(dx) This is a probability measure,in keeping with (9.el) for certainly not obvious at this s t age that Still the problem of showing that
A-;
1 v,l
y" 1/12, but it is
is infinitely divisible.
decreases is now reduced to showing
that V -+ l V, 1 / (\1+1)
• 1S
d ecreas mg • ,
V>
- I.
The argument is completed by appealing to the result of Getoor and Sharpe
05J,
(B.i) where (Rt,t
that 1 V, 1
(A(t),t 0)
0)
is the
distribution of
A{L )/2y2, v> - I, Y
is the additive functional of the Bessel process
defined by
to
339
(9.j)
A(t) • Jt I(R
o
< y)ds,
s-
the final touch being the fact that for (9.kV
lJ
-+
plJ/lJ is decreasing on Y
y > 0,
!LY-'
lJ > 0,
which is obvious after using (4.9) to write (9.k2)
for an arbitrary fixed
V
>
o.
Turning now to the production of of
V, K the (9.
u
distribution of V -+
The steps (9.ml)
nV
Kv,e,
nV
let
be the Levy measure
Ly/2y2. We want to show that
is decreasing,
V > 0.
to obtain (9.i) give this time e
-ax
nV (dx)
•
, V
>
o.
The substitution v = lJ + 1 now reveals that (9.m2)
but the appearance of
lJ + 1 on the right is most frustrating. Indeed,
shall establish below an analogue of (9.i), namely (9.n)
where (9.0)
is the (B(t),t > 0) B(t) •
plJ y
distribution of
B(L
2
Y
)/2y , lJ > 0,
is the additive functional t
r
JO
s > y)d/!.
1 (R
and in view of (9.m2) and (9.k). it follows that (9.l) holds for
V
> I.
we
340
This gives the existence of be obtained by letting
Kv,a
for
v> 1, a > 0, and
K1,a
can
v + I, but we are cheated of the result for
0 0, with a representation of this f.unction as the Laplace
transform of
2x nV(dx), a measure which has infinite mass for
in view of (9.e2). Ismail [ZtJ showed that actually c.m. for all real for v < below.
e+
V and
e
K,,_a(v'a) /
0 < '11
0
spent
a result
and due originally to Cieselski and
3,4, .... Curiously, the companion with e = I do not seem to combine to yield
(d-2)/2, d
identities (9.n) and (9.0) for K such an attractive result.
Ismail and Kelker [26J give formulae for the densities of
in terms of the Bessel functions
and
We come now to the proof of (9.n). We shall establish the following result, which encompasses (9.i), (9.n) and the joint Laplace transform of the BESy(V)
before
for
x = y
by
spent below and above
giving y
by
Ly' We use the notation of (9.j) and (9.0).
Proposition (9.2) : Por
V
> 0
V Ey exp{-CaA(Ly )+SB(Ly »/ 2y 2}
z»
=
IV-I
Note : To recover (9.i) and (9.n) put either
I (,Ta) K \)
\!
Cia) K)v13)+Ii3KV-1
a or S equal to zero and
use the recurrence formulae (13.4). To recover
x = y, put a = a
for
and use the formula (13. S) for the Hronskian. Proof: Fix v > 0 and put
Since
f(O)
calculate
= Eo
f(x)
= ExV exp{-
(oA(L ) +SB(L y
y
»/ 2y2}.
exp(- aT / 2y2)f(y), it suffices in view of (9.a2) to y
f(O). But by remark (4.2) (ii) and Williams time reversal
theorem (3.3), or by the result of Sharpe [4IJ, f(O) = g(y), 't-7here V g(x) = E- exp{-(aA(T ) + aB(T 2y2} x
0
-v
where Px governs a BESx(-V) zero. Now g is a solution of G_vg where
G_
= ag
on
0
»/
process up to the time
(O,y) ; G_vg = ag
v is the generator of
BES(-v).
on
To when it hits
(y,oo),
342
The determination of
g
is now completed in the manner of
[is]
Section 8,
the constants in the general solution being determined by the boundary conditions g(O) = I ; g(y-) .. g(y+) ; s ' (y-) .. g' (y+).
It is also possible to derive (9.2) from the special cases (9.i) and mentioned earlier, using the independence of the excursions above and below
y - see the end of this section.
We address now the question of complete
of the ratios
n.n) in the Introduction. Our description of the positive measures on [0,(0) 2 which have these ratios as Laplace transforms is based on the limit as
0
of the measures
noted, increase as
decreases.
(9.3) :
.!.
(i)
u
Y
Z
y
y
y > O. Thdn for
exp(t /
does not depend on
This
--
=2
_,which, as we have already
> 0, end
Z
0, y
C(L )Jd;f M (Z) y y
M
y
is a
infinite total mass. Moreover, for each
cr-finite measure on 0 > 0
Y
-
_measurable
with
I (oy) K (oy) Eo,a Z exp(t 0 2 L ) ' y 0 0 y
(ii)
1'1 (Z)
(iii)
M (Z) .. lim.!. y u
Y
on
and
:
Y
The identities (i) and (ii) follow immediately from the formulae
of (4.8) and (4.9) for change of law on (3.xb), using (13.2).
_' and (iii) follmqs like y
343
the identity (ii) shows that My here is BES (0) in (3.8). Thus the descripy in (3.7) applies, where from (7.a), and (2.h) we find In view of
Rerml'k (9.4)
identical to the My tion of M
y
associated with
My(LyEdt)
m
dt
in keeping with (3.y) • The measure M can also be described as the image of the measure M y
associated with
BM
in (3.8) after the spoce transformation and random
time change described at the end of section 7.
Pl'oposition
Z > 0 be
(9.4) : Let
!L _measurable. The function y
(i)
is the Laplace transform of a positive measure on -I
f(a,V) - V
Let
V
E exp(-aZ). y
Then the function (ii)
(a,v)
f(a,lV)
is the Laplace transform of a pod ti ve measure on If
Z - A(Ly )
for an additive functional
[0,00) 2•
A, then, for each
V > 0, the
function
(iii)
a
Vf(a,v) (=
(exp -aZ»
is the Laplace transform of an infinitely divisible law on same is true of the function (iv)
v
f(a,lV) / f(a,O+),
for each a > 0 such that
f(a,O+)
0
except in the trivial case when
At
=0
f(a,O+)
0,
C(L)
P
v >
O. the law
is infinitely divisible.
y
has
y
probability
V
under
infinitely divisible law under the
QV(F) Y
M [exp (-az -
Y
M
y
+v
[exp(-az - -} v
2 2
C(I,
» ;
FJ
Y
C(L
y
»J
and (iv) follows because the collection of
divisible
on
is closed under weak
Remarks (9.5) : (i)
Suppose that for each
v, Z has a right-continuous density
gV(z), z > O. Then by approximating
gV(z) by
pV(z < Z < z + £)/£, and
using (9.4) (i) with the indicator of the event of
Z. we Bee that for each
(z.:::. Z .:::. z + £)
z > 0, the function
(9.pl)
is the pointwise limit of c.m. functions, hence itself c.m ••
instead
R+
345
(ii) The proof of (ii) shows that f(a,v) where
= v-I
EV exp(-aZ) Y
is a density for the
h (0)
a
Assuming that for each
a
= ro
Jo exp(-
M (dw)e y
-az(w)
1 -2
v
2t)
h (t)dt, a
distribution of
this density is right-continuous in
sort of argument used in the last remark shows that for each
G(L ). y
t, the same t > 0, the
function (9.p2) is e.m ..
Exq!'p,zes
(i)
(9. 6)
Take
Z
= Ly
f(a,v)
in (9.4). From
= 2I v (1Za
y) K
v
(12a y),
r > 0
hence from (9.4) (ii), for each
we have
the function
(9.ql)
a result which ue already know from (7 .f). This cotapLemenr s the fact, noted in [51], that as a consequence of (2.j) each of Hartman's functions (l.a), (l.b), (l.c) and t l cd) becomes completely
(a,v)
monotone
after the substitution
of an extra factor of
/.i
A v/Z and the
in the 2c8llilsnt of each Bessel function. The
infini tely divisible laws on tho: line arising from (9.4) (iii) and (iv) have already appeared in
and (7.f). From (9.pl) we recover the c.m.
of the first Hartman function (l.a)while the conclusion of (9.p2) seems very complicated. (ii) Take pV y
Z
= A(Ly ) '
distribution of
Z/ 2y2
the total time spent below is
the c.m. property of the first Taking
y
show that
= 1//2,
lV-I,I, and (9.4) (ii) and (9.a2) yield of the
Ismail and Kelker
Z has density V
g (z) - 4V Ev_l(z),
y. From (9.i) the
z > 0
functions in (l.n). ; theorem 1.9 and formula (4.15)
346
where for
II
- I, 00
exp ( - J.2
(B.q2)
ll,n
) z.
: n =
(j 11,n
I ,Z, ••• )
being the increasing sequence of positive zeros of the Bessel function
J . From (B.pl) we learn that for each
of the first kind
11
function (B.q3)
\I -+
LIv_1 (a)
z > 0
the
is
hence also (see (9.7) below) (B.q3')
(iii) Take Z a B(L ), the time spent above y before L. From (B.n) y y and (9.4) (ii) we obtain the c.m. property of the second of the functions in O. n)
,
have, for
On the other hand. following Grosswald
1\> 0
the function
y
we
k\l(z)dz,
and from (9.p]) we find that for each
Taking
[j4],
> - I,
\I
the
pV y
is therefore
2VK,,(Z). v
(9.q5)
Ismail also showed that
where 1 m (r) .. -
zn
x
roo 0
ds e -st/Z
which identifies the function Thus for each (9.q6)
Im{K . ,-(x) K) . ,-(x)}
mlei(t)/I 0,
n" 1.2 •••••
..
(a,v) ....
(9.1'1)
(lei) n
and for each (9.1'2)
a> 0,
Ke+1\i + n (I0) u
time with rate
(7.b), and the fact that the jumps to 00 after an exponential
-
v/y, one finds that for
V E [exp(-aT ) y u
; T < u
(0)
=
v >0
exp [-u/gVa(y ,y)J
where
This is a special case of a well known formula which holds for any Markov process with nice enough transition function - see e.g. Getoor
04J,
(iv)
If
formulae ('1.9) and (7.15). exp(-w(a»
is the Laplace transform in a
[0,(0), then W(O) • 0, and
divisible probability law on
of an infinitely w(a)
has a c.m.
derivative (9. iil )
where
c
tions, and
d da w(a)
=c
+
[
0 e
-at tA(dt),
is a positive constant which will always be zero
our apFlica-
A is the usual Levy meascre satisfying A(t,oo)
O. It fo l Lows by the Criterion 2 of Feller [10], XIII.4, (9. t2)
if
f
is c.m. then so is
w(a)/a
=c
+
J:
e-
at
Applying these observations to the that for
v >-
the function of
1
that
f(¢),
and an intef,ration by parts shows also that (9.t3)
for all
w(a)/a
is c.m. with
A(t,ro)dt.
W's in a
we leRrn fuom (9.t1)
(with the substitution
a = y
(9.t4)
v
and similarly for (9.t4')
2y
> 0
v da Wy(o,a) = - I +
-I d
2
is c.m •.
12 0, \) > - I, -
to the result of SUlcz
which is
\)
[44]
for
J'\). From (9.t4') we
K\) which we have not seen in the
obtain a companion inequality for literature
(9.t5')
o
KV-I(a)
a > 0, \) > I.
-
The Laplace transforms (9.t4) and (9.t4')
determine the corresponding
Levy measures by (9.tV, but the Levy measures in question are specified much more
by the alternative formula
a
y
by
in (9.s7) and taking
Laplace transform of
(9.q4)
v-I
V 1 ,1
y
Indeed after dividing
we recognise a multiple of the
in (9.al) and a multiple of the function in
instead of
v, Thus the Levy measure BES (v)
bution of the time spent by local time at
= 1//:2
first reaches
y
belnw
I (given
y
Av y
for the distri-
before the tiIDn
'] < 00)
T
I
that
is given by
(9. t6)
where
Lv is defined in (9.q2), and the corresponding Levy measure for
time spent above (9. t7)
where
k
y
v
is
ny 0,
v>
v is defined below (9.q4). Adding the two
Le7Y measure for
- I,
gives the
'I'
Recalling that the inverse local time process at an exponentially distributed
with rate
(LU'U
0)
jumps to
00
vly, for \) > 0 these
results can be re-expressed as follows, without reference to local time.
352 For fixed
y > 0
process
(R
s
let
N
> 0) -
.3
N
and
+
+
t > 0, let
of duration at least
above
y
: R
= y,
=1/--{s
s
be the number of excursions of the
N
R
s+u > Y
t,
d,
0 < u
0,
and as in (9.4), let
= V-I where
(A(t),t
0)
V
E exp{-aA(L )}, 0, V > 0, Y Y is an additive functional. Then for
> 0,
0,
V > 0
(9.M)
u) -
(i Proof: For
+
t
v
2
Tu
0, let ¢u (c ,») = Eu
2 C(L )} Y
1
Y
y
=
by (4.8). is given by (9.s5)
The corresponding exponent
=
as
-
and this implies (9.w) for
> O. The extension to the case
=0
is
justified by (9.88).
The formula (9.w) shows that, provided
(i)
zero, the def i ni,tion of > 0, v> 0, with (9.w) holds for all
f (a, v)
= 00
Z
a
iff
A(L ), the measure in (9.4) (ii) is y
is not identically
= V
=0
(see note below (9.4). Then
O.
(ii) By remark (3.9), formula (9.w) for of the two-dimensional
At
may be extended continuously to all
subordinator
-I Y
a
0
shows that for
times the potential measure
355
10. THE von MISES DISTRIBUTION IS NOT AUNIQUE MIXTURE OF WRAPPED NORMALS.
parameter
8 = 0, specified as in (5.b) by
k > O. centered at
[21T
vM(k,d8) -
(10. a)
I
S , with
distribution vM(k) on the circle
Consider the
1 (k)r
1
0
exp(k cos 8),
0
e
0,
ao.» where
vM(k) =
wN(v)
J:
wN(v)
is the distribution modulo
random variable with mean 0 on
(0,00)
and variance
21T
of a normally distributed
v. and
n
is the distribution r with Laplace transform (l.a). As noted below Theorem (1.1), this
result admits a direct probabilistic expression in terms of a Brownian 2 motion in R with drift 0 - k started at the origin. For vM(k) is then the distribution of distribution of
a
(see (5.2), while wN(v) is the conditional Tt BTl given C(Tx'oo) = v by Theorem (1.1), and is
the distribution of
C(Tx'oo)
by (4.i).
interpretation of (lO.b)
but in terms of conditioned processes obtained from can be read from (l.e). and Watson raised, the mixing measure
n
k
in (lO.b)
EM with no drift the question of whether
is unique. We answer this question here
in the negative : for each k.
n is not even an point of the k convex set of possible mixing measures. To see this, observe that is determined by its sequence of Fourier
coefficients, the [
where
Fk 1
o
n th of which can be expressed using (10.b)
nk{dv) e
is the law on
-
1
2 • fl
n v
0
as
2
n Fk(dx)x,
[O,IJ obtained from nk on
[0,00)
by the map
v ... e- TV. Thus n is unique if and only if there exists no other k distribution on [0.1] with the same n 2-moments as F k•
356
According to a facous theoreo of
°
for a sequence of non-negative
integers 2 nCO) < n(l) 0,
[O,IJ with thG same m(i)-moments as F.
ProOf: According to a variant of Uuntz's theorera, the functions xn(O), xn(l) ...
are complete in
L2[O,IJ
iff
Bya change of variable, the same is true in
= I,
n(O)
n(i)
= m(i-I)+I.
nal to both
and
it be zero on
(£,i], and
¢ in L2[0,£J
there exists
xm(i)+1
for all put
El/n(i) =
2 L [0,e] .
J:
i , Extend
g(x)
=
Thus, by taking
to
¢
00.
which is orthogo[O,i] by letting
q,(y)dy. By an integration by
parts (m(i)+l) Thus for any both
[0. iJ
0 >
° with
(f + og) (x) dx with the sane
probabilities is
and
0
2
xm(i)+l q,(x)dx
£/sup(-'a,b), where
(f _. og) (x) dx
ra(i)-monents as
has a continuous density
f
k
= O.
a = inf g. b '" sup g,
are probability measures on
f. Since the average
F, the conclusion of
To see that the Lemma applies to F k
= -
xm(i)g(x)dx
of these two
Leoma is evident.
F '" F
such that
k
for each fk(O+)
=
k, we argue that
00.
This is an irJmediate consequence of the formula (5.7) (ii) of [51J continuous density
h
k
of
n,let• which
c(k) u- 3/ 2 for some constant
c(k).
as
shows that u
+
00
for the
357
11. OTHER WRAPPINGS Let
Lx be the last time at
x
for the
radial part
(Rt,t
0) of
a m1 in R2 starting at zero with drift 0 > 0 in the direction e., o. According to Theorem (1.1) and formula (l.i). the distribution of the angle 0(L is a mixture of wrapped normal distributions wN(v) with x) mixing measure the second Hartman law with Laplace transform (l.b) fer parameter
r" ox. The n th Fourier coefficient of this distribution is
therefore
K
(r)/K (r), o n (D.!}(i). we learn that
(ll.a)
1 --2
TI
L e
and since these coefficients have a finite sum by 8(L
i ne
nEZ
has a bounded continuous density
x)
K (r)/K (r), n
0
< e
0,
359
Remarks (l1.V :
(i)
Let
be the transition density of
plane with drift 0
in the real direction. It is easy to verify that 'IT
and, put togather with circle of radius circle for
in the complex
x
-1
Ko (Iz-z' I) exp ORe(z-z')
this yields a formula for the density on the
of the equilibrium (or capacitary) measure of this
BH with drift
0 (see Getoor-Sharpe
[t6J
or Chung [4J). The
theory of last exit distributions as developed on these papers can now be applied quite routinely to obtain formulae for the joint density of the time and place of the last exit from the circle of radius
x
with drift
rather than
z
= o.
0
started at an arbitrary initial position
z
for
EM
Though quite complicated these formulae are as explicit as (11.aJ. these formulae gives a means of calculating the hitting
probability for the circle which seems simpler than a direct attack on the
eM
formula and
of Wendel
(47].
(ii) Everything above can be extended to integer dimensions and perhaps even to real dinensions in the manner of Kent results do not appear to deserve space.
BU,
d
based
2,
but the
360
12. CONCLUDING We mention first a characterization of certain processes which admit a skew-product decomposition
to that of
Let
n = COR+JR ). d > 2. We
us work on the canonical space prdbabilities
with no drift in
d
P on n satisfying
#ypothesis (12.1) : There exists
b
R+ x
n .... R d • uniformly bound,::;cl and.
(F )-predictable such that =t
b(s,w)ds.
= a(xs'c
Here of course
e
t
Sd-I
'
s
t). We put
Xt
and let
FToposition (12.2)
Under Hypothesis (12.1),
(X
t
)
admi t s a skew-product,
representation
(12. al ) where
is a
0)
in Sd-I
increasing process adapted to
Io t
(12.a2)
and there exists
r
(R) =t
independent of
if and only if both
b(t.w)
t
is an
2
cls/R ,
s
R+ x
n .... n, a bounded predictable scalar valued
process, such that
(ie.t»
(R ) and (At)
= r(t.w)
8(t,w)
a.e.
dt dP.
361
Proof: As is easily verified, if
of refining partitions of a
d-r I
BM in S
= Son
0
with
n
+
00.
n(
lim sup n
, then, denoting the
in probability as
n
< s) < ••• < sm s)
=
S
0, and
-
k
d
norm in
is a sequence
by
f'\'
Using the fact that under(12.al) the laws
(Wiener Qeasure starting at
are equivalent on
x)
F
"'t
is
for each
P and t > 0,
one deduces fr041 this that the representation (12.al) holds iff the clock is of the faniJ.iar Browrri an form t ie.az); Let version of the density of p(FIR)
=t
=
P with respect to
(L
now be a (continuous)
t)
VI • Now (l 2. al ) holds iff
x
> 0,
=t
and it is iaJediate that this is in turn equivalent to Lt
-is
R =t -n1"",;.:ec::a;.:s.:;u;.:r;::ab::..l::..e::."'/8, and
2
.jJKV = WKV
if
b
b.
a
t =
equation
364
Proof: Fix (O,b)
and let (b,OO) ,
and on
G W
a
where G
a
is the
BES(O)
W(a)
be the expectation in (12.k). Then on
W solves
[1. Z
a
Z
+
a
+
1. 6 2JW, 2
2 1 d 1 d "'2"--2 + - -
cla
2a da
generator. A change of variables now reveals that
W(a) = (Z6a)_1/zY(28a)
where
y
is a solution of vlliittaker's equation (12.j).
and (12.k) results after the usual consideration of boundary conditions. The above proposition provides probabilistic interpretations and ex!:rmsions of soxe of the resul ts of Har trnan [Z:U (see in particular (4.13) and (4. 14) of [22J). Also, if
[3CO
motion of Kendall polar drift
c
P
gove rns the pole-seeking Brown i an
with generator (12.fJ, in
starting at a point
xeY'z
V!ith
d
=2
Ixl '"
dimensions with a
and angle
e=
after using first the skew-product (12.1), then the change of Vnv fa rmu 11'. (12.h), and finally (12.kJ, we obtain U2.1)
where the In the case
(
E exp i\.l8(T
b
) _ NT b ) u.
are as in (12.k), c < 0
and
C
D)l /z £
= e ( b •.a)(:"-a
(ZYa) •
DOW
y
=
(c 2 + za)l/Z,K
= -c/2y.
a > b , this is the formula (32) of Kendall [3iB.
0,
365
13. APPEN0IX. FORMULAE FOR BESSEL FUNCTIONS We record here for the reader1s convenience Bessel functions
Kv
Iv and
formulae and definitions of Stegun [IJ
I
-n (:i!)
=
I (z) ;
n
ASyEptotics. As
K (z)
(13.3)
Iv and Kv may be found in Abramowitz and
indices. For integer
(13.1)
(13.2)
[8J.
or Erdelyi
'V -}
z
r (v) C-}
K_" (z) v
n
0, v
fixCG
z) -v,
v > O.
Wronskian
Derivatives and
uZ
L
v (z)
and real
l\> (z) •
W{K (z),I (z)} V v (13.4)
those formulae for the
which have been used in the paper. TI,ese
recurrences For
v
366
REFEREf!CES
[2]
M. ABRAMOWITZ, I. STEGUN
Handbook of Mathematical Functions. New York - Dover - 1970.
L. BONDESSON
A general result on infinite divisibility, The Annals of Probability, nO 6, I (1979), 965-979.
L. BRED1AN
Probability. Addison - Wesley, Reading, Mass. 1968.
K.L. CHUNG
[5J
Probabilistic approach in potential theory to the equilibrium problem. Ann. lust. Fourier, Grenoble, 23, nO 3, 313-322, 1973. for Z. CIESIELSKI,S.J. TAYLOR: First passage times and sojourn in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103, 434-450, 1962.
[qJ
J.L.
S.F.
Conditional Bro.mian motion and the boundary of functions. Bull. Soc. Hath. Fran.ce. 85 (1957), 431-458. constraints : I. Proc. Phys. Soc.,
with topological 513-519 (1967).
[8J
A. ERDELYI, and nl
Tables of Integral Transforms, vol. I, Mc Graw - Hill, New York, 1953.
t9J
w.
FELLER
InfinitelY divisible distributions and Bessel functions associated with random walks, J. Appl. vol 14, 4 (1966), 864-875. -
[IOJ
w.
FELLER
An Introduction to Probability Theory and its Applications, vol II, Wiley, New York, 1966.
III]
D.
Brownian and diffusion. San Francisco - Cambridge - London Holden-Day (1971).
367
[13]
[17J
A.R.
Representation of an diffusion as a skew-prodtict, Zeitsctlrift fur Wahr., 1 (J 963), 359-378.
R.K. GETOOR
The brownian escape process, The Annals of Probability, n b 5, I (1979), 864-867.
R.K. GETOOR
Excursions of process. Annals of Probability, I, nO 2, 244-266 (1979).
R.K. GErOOR, U.J. SP.ARPE
Excursions of brownian motion and Bessel processes, Zeitschrift ftr Wahr., (1979), 83-106.
R.K. G"RTOOR, r1.J. SElu'U'E
Last exit times and additive functionals, The Annals of Probability, vol. 1,4 (1973), 550-569. -
L. GORDON, H. mJDSON
A characterization of the distribution. Ann. Stat. l, 813-814, 1977.
E. GROSSFALD
The t-distribution of any degree of freedom is infinitely Zeitschrift ftr Wahr., 36 (1976), 103-]09.
J • H. IW1HERSLEY
On the statistical loss of long - period comets from the solar system II. Proceedings of the 4 t h Berkeley Symp. on Stat. and Probability. (1960) Volume III : Astronomy and Physics.
P. HA.l1'W,AN, G. S. WATSON
"Normal" distribution functions on spberes and the l':odified Bessel functions, Ann. Probability, ! (1974), 593-607.
P. HARTHAN
monotone families of solutions of n th order linear. differential equations and infinitely divisLble distributions, Ann. Scuola Norm. Sup. Pisa, IV, vol III (1976), 267-287.
368 P. P.ARTIftili
[D]
if.E. ISJ:1AIL
Uniqueness of principal values, CODplete uonotonicity of Logarith8ic of principal and Oscillation theorens. Hath. Anna Ler- 241, 257-281, ] 979. Bessel Functions and the infinite divisibility
Of the Student t-distriaution. Annals of Proba,
2,
n° 4, 582-585, 1977.
[24J
H.E. ISHAIL
ITItegral representations and Corplete nonotonicity of varicus quotients of Ressel functions. Canadian J. XXIX. 1198-1207, 1977.
[25J
H.E. ISHAIL, D.H. KELKER
The. Bessel Polvnol'.::ials .:md the "tudent
[26]
t-·distribution. J. Math. Anal.
M.E. ISMAIL, D.H. KELKER
K. ITO
I.
1976.
Special Functions, Stieltjes and Irrfinite divisibility. Sias J. Anal. vol la, nO 5, 884-901, 1979. Poisson point processes attached to proceSS2S. c0 th Ber.e k 1 ey *T'
We can do this construction of probability measures simultaneously for observables AI' A ... An that commute in the Z' . strong sense that the groups e , k = 1,2, ••• N conmut.e •
= Es+t •
We also note that W(t)E W-1(t) s
We can now evaluate the Schwinger functions by using the limit (3), the Feynman-Kac formula (4) and the Markov property,
(5).
From (3), we have
lim
T-+m
(6) This is valid provided T > s1 > s2 > ..• > sn > -T.
From (4), we
insert EO exp[- rSj-Sj+l V(Q(s) )dslw(s. - s , l)E J J J+ 0
J
o
-(s.-s. l)H for the semigroup e J J+ , j = 1, ••• n-l, and a factor Eoexp[-J:+Sn V(Q(S»dS]W(T+Sn)E
o
for e-(T+Sn)H, acting on 1.
The
tezms W(t+s )E can be ignored, as E 1 = 1, W(S) 1 = 1. The n O T+s 0 remaining expression exp[-L n V(Q(S»dS] lies in the future, as T+S > O. n
Also EOQ(O)E
O
=
EOQ(O).
Putting these remarks
together, the last few tems in the numerator in (6) become
384
e
-(5
n-l
-s lH -(T+s lH n Q(O)e n 1 S
[J
EOexp - 0
n-l
-5
n
V(Q(s» dS]W(Sn_l
T+S x exp[- 0
f
n V(Q(S»dS]l.
Now take W(Sn_l - Sn) of 1.
to the right, and use the invariance
The term becomes Eoexpl-fsn-l-Sn V(Q(S»dS]E -s Q(sn_l s n-l n 0
L
[J
x exp _
T+S 5
n-
n-l
The terms on the left of E
5
)
x
n
1
-5
n
-s are in the past of sn - sn_l' n-l n those on the right are in the future of sn - Sn_l. Hence, by s
(5), we may remove the conditional expectation E Collecting up terms,
s_ s n n-l (which all commute), this part of the
numerator comes to T+ S
[J
EOexp _
O
n-
1
V(Q(s»dslQ(s
'J
n-
1 -
5
n
)l
(7)
Proceeding in the same way with the next factor in the numerator, namely
and combining with (7), we get
We may proceed to the end, and use translation invariance to obtain
385
I
has unit modulus.
This was suggested
by Nelson, and it gives a quick proof of the "diamagnetic inequalities" first proved by Simon using other methods.
See [4J.
The F-K-I formula suggests a general way to quantize systems
in an electromagnetic field:
we just multiply the Gibbs factor
G in the functional integral by
This has been
used by physicists to quantize a Syste: on any Riemanifold if
Jt
paths
has a boundary
aJf"
that do not hit
;
we restrict the functional integral to
aJ1J
in the time interval [0, t.},
We
would then need to show, post hoc, that the quantum theory thus obtained (by projection to the time-zero space) was the one we sought, e.g. that the Hamiltonian is the covariant Laplacian on
Jt
with Dirichlet boundary conditions on
usually omitted.
aJ(,.
This step is
390 §6.
Non-abelian gauge fields -i
The famous factor e
is a (l-dimensional) unitary
representation of a random element (i.e. an element depending on of the "gauge group" U (1)
•
In this form it has a natural
generalization to other, non-abelian, Lie groups. Let G, the gauge group, be a given Lie group, and let V be a continuous unitary representation of G on a complex Hilbert space L, with dim L
We say that a gauge field is given, if
we are given the following. continuous path
W
To each pair x, y in Em, and each
from x to y, is given a group element g(x, y; w)
obeying 1-
g(x, x', 0) = IG' the identity of G.
2.
g(x, y;
3.
g(x, y; !:'Ug(y,
=
[g(y,
Xi
=
Zi
g(x, z;
We require the map g to be jointly continuous in x, y and measurable in .!e.. The space of wave-functions for a "multiplet" of
particles
related by the symmetry group G is taken to be the Hilbert space K = L @ L2QRm); a wave-function in K is, equivalently, described by a column vector {WO
j
Here, j is the unit vector along x., and w is the straight line J
between x and x + Aj.
The covariant derivative is
lim A-1{V(X, x + Aj;
+ Ajl -
A4()
d dX
( - - iA. (x) J
j
(x) ,
According to the theory of path-ordered integrals [8J we can recover V from A by
vtx, y; for each smooth
(yl = Pe
iVV.dw x-
(x)
We hope to make sense of this relation
for all paths in Q, and would expect the correct form to involve an ordered Stratanovich integral. The energy operator for a SchrBdinger particle in a given gauge potential A is related to the covariant Laplacian t> 1
"2
H(Al
m
LV .• V. j=l J J
We can now pursue the Euclidean method, as in §5, to obtain
This shows that the semigroup is the expectation of a random unitary operator. e
X+Y
Again, let us use (12l X X 2 Y 2
=e e e
for infinitesimal random operators.
Then
thus:
392
e
'i/.l.iw
N
-H(A)t
IT Ee k=l
-
-
-k
kt
where
""N'
Thus e
-H (A) t
-
_
l
- E e
2.. 0!!l.1 2.. .e
.
•. , e
2..
e
e
. ... e
e
e
J
if the limit exists. Let us move all the translation operators to the right. build up to
They
We are left with a path ordered product of
unitaries, N [
IT e k=l
•
which has the Stratanovich character.
Progress towards proof of
convergence of products like this is reported by Parthasarthy and Sinha [9J.
I would like to thank Miss K. Anderson for her careful typing of the manuscript.
393 References [lJ
R.F. Streater and A.S. Wightman, PCT, Spin and Statistics and All That.
[2J
Benjamin/Cummings 2nd Ed. N.Y. 1978.
G.G. Emch, Algebraic Methods in statistical Mechanics and Quantum Field Theory.
[3J
Wiley-Interscience, 1972.
K. Osterwalder and R. Schrader, Axioms for Euclidean Greens functions.
Cornmun. Math. Phys.
83
(1973); and
il,
281
(1975) • [4J
B. Simon, Functional Integration and Quantum Physics. (Academic Press, 1979).
[5J
J. Frohlich, The reconstruction of quantum fields from Euclidean Green's functions at arbitrary temperatures; Relv. Phys. Acta 48, 355-363 (1975).
[6J
A. Klein and L.J. Landau, Stochastic Processes associated with KMS states.
Preprint, University of California,
Irvine. [7J
S. Albeverio and R. Roegh-Krohn, Uniqueness of the Physical Vacuum and the Wightman Functions in the Infinite Volume Limit for Some non Polynomial Interactions.
Cornmun. Math.
Phys. 30, 171-200 (1973). [8J
J. Dollard and C. Friedman, On strong product integration. Jour. Funct. Anal. 28, 309 (1978).
[9J
K. Parthasarathy and B. Sinha, a Random Kato-Trotter product formula.
Preprint, India Statistical Institute, New Delhi.
The Malliavin Calculus and its Applications Daniel W. Stroock
*This
work was partially supported by N.S,F. Grant MCS 77-14881.
395
Lecture #);;
2.
Statement of the General Problem: Let
function under
(E ,:'1, P)
be a probabi lity space,
l t: E ... R
\lot
,let
= p. t -1
Given an
:'! - measurable
be the distribution of
t
The central problem addressed in these lectures will be the
P
development of techniques to answer the question:
when is
\lot
continuous and, if it is, what can be said about its density? we will be studying this question only when
(E,:,!,P)
absolutely Actually,
is Wiener space;
but in the hope that the underlying ideas will be clearer in a more general setting, we will begin with an
abstract treatment.
To begin with, consid er how an analyst
would attack this problem.
An analyst confronted with the problem of showing that a measure
IJo
has a density would attempt to obtain estimates of the form:
on (0.1)
where
ql
(n) = Dnl\'
uni form norm. 'I'
E
th n-
is the
derivative of
Indeed, suppose that Taking
'l'
S
0
n ,;; N
In particular, if theory,
\lo
l
and
N
E R
2
has a density f
(m)
f E Cb(R 1
o ,;;
denotes the n ,;; N and
C n
,where
,then
(x) = 2n
II '\\u
, we would then have:
IJtpt)dlJo\ for all
and
holds for
(0,1)
=
(x)
0
i
(\i1,,r'lt)
Definition:
(1.12) Remark:
'I!)
0
i=l
'l!» . .
Since
n
=
'I!)
0
E ilCJ: 1)
E il
E
:
p
n LP(P)
X
p
(1.7)
X2
,
= il 2 = il(£)
.
Also, each
can be made into a Banach space with norm
(By Theorem (1.7)
\\I'\lI
J
x2 valent to the graph{J'> norm.) normed
1\\ ·\\\X
for each
p
satisfies:
\F(x)\
and
\
max
Id
P
2
2 < p < m
function satisfying
Proof:
2
n
C (R) b
dominated by
lit
n ... ,t ) EX, n p
=
If
l)/2
is given by the right hand side of
1
In particular, if
Finally, if
Cl(l+ !x\2)(cr
oXi
2 ( 2)/2
n
F E C (R )
\li(x)\
2
.1:
2
and suppose
C (l+\x\ ) a
(x)\
oXioxj
E IJ / p a
F'
(1.2 )
from
X 2
space in which convergence corresponds to convergence in
Corollary (1.13):
then
on
and I
q < p/2
E L3p/p2q(p)
• and I
then
p
t EX p
E il q
The first part is proved by approximating and observing that
is a nonnegative
\\right hand side of
F
with functions is
403
The second assertion is just a special case of the first. the last part can be easi ly proved by considering F (x) .. E
1/ (x
Z
Z
+ E ) 1/ 2
.
F 01 E
Finally,
where
The desired result follows upon letting
E
I 0
Q.E.D. We are at last ready
see how all of this machinery can be put
to
to work on the problem which we posed at the beginning. Suppose that
1/(1,1) E
n
t E K has the properties that
Given
LP(P)
P ,£«(cp 01)-1) -
.. E [
(
(t,l) E K and that
,we then have:
E
0 I)-a -1-,£(9' 0 t)]
t.t )
Hence
where
C=\\(t,1/(I,I»\\ 1
L (P)
1/2
L (P)
\\(1/(1,1) .1/(1.1»\\ 1
+ 211'u1l 2 L
2
' L (P)
1/2
11(1,1)\1 1
31\1111l x
+ 2\1 (IUI)\\ 1
L (P)
\I 1/(1,1)\\
(P)
2
2
L (P)
404
In other words, i f (1.
• E
1/(•. t)
and
•
E
then
14) (0.1)
A refinement of the technique used in section \lot
itself already implies that t E LP(R l) for 1 p < dx
shows that (1.14)
by
is absolutely continuous and that
We conclude this lecture with the reformulation of the preceding in the form which will be most useful to uS in the sequel . (l.15) Definition:
It.w E M\
t
11)
if
x(n+l)
so that:
K(n)
5
M
• define
Now use induction on
E X(n)
for
Given
[.
i)
X(l). X and
and
for
• E X(n)
E K(n) :
t E x(n+l)
to define
n" 1
(t)
.soo = tlt.H.(It.1l!> x(n)
and
:
j.(n) (It) and
Ii
have been defined. then
and
Finally. let
= n}(n)
1
Lemma (1.16):
If
(the space of
t= (tl •...• t
lit E Proof:
E
and
function which together with all their derivatives F· t E K
are slowly increasing). then and
n)
n
LP(P)
...
1/t
• then
Moreover.
E
In both cases. the proof is a straight forward inductive
argument which turns on Corollary
(1.13)
Q. E. D.
Theorem (1.17):
A = «(ti"j))lsi.jSn
ct.e.+i.i\e
and
t = (.l •...• t ) E n
6
= det
A
and set
1!...J
j=l
«A(ij»)
[A (ij) (2'!1l:t. + (W,It.) +\II(t .• A (i J
J
is the cofactor matrix of
and
For
n
l(.\II = S
where
Let
J
A
j»)]
405
(l.18 ) In particular, if
P
(l
1/tJ. E
,then
L (P)
(1.19) where
Finally, assume
;r = WI
0'"
0
"n
(l
1/tJ. E
_an
0'1
LP(P)
Then for
and for ql
0'
=
define
n
ex>
E C (R ) b
(1.20) Proof:
The proof of
to arrive at (1.20)
(1.14)
(1.18) Once
is precisely like the argument used
(1.18)
ha 5 been proved,
(1.1 9)
and
follow easily. Q. E. D.
Corollary (1.21): 1/tJ. E
n
LP(P)
Let everything be as in Theorem
,then the distribution
IJot
of
,
under
absolutely continuous with respect to Lebesgue measure on dlJo'
d;Z" E
(1.17) P n R
If is and
n (R )
Proof:
Once one has
(1.20)
,one obtains ,
simply by taking
=
1
01 1 P < co
contraction semi-group S(Pl F= S F 'T T
(1.3)
will be the extention of
But, because of Lemma
(2.7), this identification
boils to checking that S H
'f a 'f
for all
2
- H
ex L
(n 1/2
-t
2
o H
) LJ
( __ tx
kE[aJ
a EA
X
oH
----.a.)
2 - k OX
oX k
k
a fact which is easy to derive.
We have now shown that
is a s yuunc t r i.c diffusion operator on
J:
Of course, we did this under the assumption that
d
=
1
but as we said at the beginning of this present section, there are no serious obstacles preventing us from doing the same thing for 2.10 Example:
Before going on, let
US
indulge in the following
ridiculous exercise; to show that the distribution of admi ts a
t > 0
- dens ity for each
2
6 (t) ,. But
t
S 6(s)d6(s)
°
2
J: (6 (t )
In particular, that
E Z2 and
F E C7CR1)
=
2S t 6(s)d6(s) o
E Zo
t
6(t)
under
\ll
10 this end, note that since
Also, by
I6(t) ,. -1/2 6(t)
d > 1
formula:
+ t
,and so
t 2 -2 S 6(s)d6(s) = t - 6 (t )
o
(6(t),6(t)i = tHence, Then for
B(t) E KCo:»
1) cP E C;CR
[CP'C6Ct»FC6(t»] = 1 \ll [('1'C8Ct»,6Ct)i F(8Ct»]
Now suppose
414
=
+ 1 IJJ
..
=
1 IJJ
(e(t)F(e (t»)] 1 til [(a(t»(a(t)F(e(t»)]
IJJ [q>(a(t»(e(t)F(e(t»)] -E [q>(e(t»F'(e(t»]
TI,at is: IJJ IJJ E [q>'(e(t»F(a(t») .. E [ql(e(t»(MtF)(a(t»)
where
M
t
F(x) ..
- F' (x)
t
Since it is clear that
MPl t
we conclude that for all
and so the distribution of
n
, and so
is an
th
n-
order polynomial for each
t
'>
D ,
D
e(t)
under
\JJ
has a
- density.
Without too much trouble, it is even possible to check that -
n/2
lim t / C (t) < tiD n e(t) as tiD
and thereby get estimates on the distribution of
Of course it is fair to ask whether we could have possibly developed the Malliavin machinery without knowing ahead of time that
a(t)
has as
nice a distribution as there ever was, but the preceding exercise gives the flavor of the applications which we have in mind. Malliavin calculus and Stochastic Integral Equations: Until the end of this section we will again restrict ourselves to
415
Once again this is a matter of convenience and is not done
d = 1
d > 1
because when
=
d
presents any essential difficulties not encountered
1 OI.e E il(L)
To begin with. suppose
II
OI·(a(t+h)
II E II (L)
+ eh
and then compute As for
are
where
h > 0
U
Clearly
OICa(t+h) -aCt»
B -measurable and let t
We want to show that eh
presents no problem and
• note that if
01
E Zn
then 01 =
S
fda (n)
(:,n
for some
2
f E L ({:, n ( t »
(r eca Ll, that
= \.(t 1 •.•. ,t n)
ERn
= S fda ClOl-1 )
(aCt+h) -
01-
(:,n(t)
(:,n
Thus, if
CJ
E Zn
is
B
t
-measurable, then
OI·(e(t+h) -aCt»
E Zn+l
and so ! COl'
n+1 --2- OI'(a(t+h) -aCt»
(aCt + h) - a (t ) ) )
=
(L0I-1/201)- (a(t+h)
Fr om here it is an easy matter to conclude in general that for measurable (3 _1)
01.1'
E llC!)
h > 0
! ( 01· (a (t + h) - a (t ) )+ e h) • Starting from
and
and
B:
Xe
(3.1)
1 R
OI(a(t+h) -
t-
+ eh E ilC!) and
(L01 - 1/2 (1) • (a (t + h) - a (t » +
• one can now show that if
B
01:
C! e ) h Xe
are progressively measurable functions such
1 R
416
a(t),e(t) E :Q
°
'
Set
that
l R
be progressively measurable functions
4 ill T E[IC\\lO'(t)l\ +\I\B(t)\\\){)dt} t t 0 4 I aCs)de(s) + I e(s)ds . If we know o ill TO 4 T> 0 and that E [I dtl < ee o 1\4 t>O
c
t > 0
-t
,and
then we could conclude from:
2
(t) '"
21 t
+ I
o
2
t
0
(s j jds
that
2
t
S. o
(t ) •
+ S.
t
2
+!(a (s»)ds
o
Since
= S. t (a(s)! o +
-1/2
S. t
+O'Cs)!(O'(s»
o
2 -1/20' (s j j d s
i t follows that
(3.3)
• I
t
o
+
J:
t
0
2
(a(s),aCS»)+a (s j jd s
,
417
Of course we arrived at
(3.3)
we have not yet proved.
However, the assumptions that we made are
trivial to verify if a([ntljn)
and
(3.3)
Hence
for simple
by assuming things about
and
a(')
t z 0
e(t) = e ([ntljn) is proved for simple and
a(·)
are simple (i.e.
8(')
n Z I ).
8(')
But from
8(')
and
(3.3)
and
I
I
b: R ... R
bounded first derivatives and define (3.4 )
x(t)
= S.
t
o
,then by
+
S.
t
o
:2:
t
0
0
by
b(x(s»ds
t
:2:
0
, and
(3.3) :
Jt 20'(x(s»(x(s),x(s»de(s)
(x(t),x(t») =
(3.5)
t
be smooth functions with
x(.)
S.
o(x(s»de(s) +
x(t) E X2
If we know that T> 0
for general
which satisfy our basic assumptions.
0: RI ... RI
Now let
(3.3)
plus standard approximation techniques
8(')
used in stochastic integration, one can justify
a(')
aCt)
,for some and
a(')
which
o
2
2
[(2b' (x t s ) +0' (x(s»)(x(s),x(s») +0 (x(s»]ds
In order to give a rigorous derivation of
(3.5)
same Picard iteration scheme that Ito used to solve
one can use the (3.4)
At each
stage one has to check that the iterate at that state has the required smoothness. jus tify at
It is then quite easy to pass to the limit and thereby (3.5)
S.
(3.6) .Lx(t) =
+
t
o
t
[1/20" (x(s»(x(s) ,xes)) + o'(x(s».Lx(s) -1/2 o(x(s»]de(s)
S. [1/2 o
Combining
At the same time, one can derive:
b" (x(s»(x(s),x(s»+b'(x(s»J:x(s)]ds
(3.4) , (3.5) ,and
(3.6)
,we see that
418
• (x(t),(x(t),x(t»,!x(t)}
again satisfies a system of
stochsstic differential equations to which we can apply the same procedure.
In this way, one can use induction to show that the Malliavin
operations may be applied aribtrarily often to
x(t)
In fact, by
being_ a little careful, one can prove the following theorem. Theorem (U):
Let
d N N cr: R -+ R 181 R
functions such that
li oX i
and
1
i
N
oX i
and
are
Given a aO-measurable
-
b: RN -+ RN be
C"'-
functions for all
E (l«",»N
let
x(·)
be
the progressively measurable function satisfying: x(t) ..
(3.8)
Then
x(t)
+
1t cr(x(s»d8(s) + 1t b(x(s»ds o
E (l«e»N for all t
e «(xi
+
t
0
Moreover, if
A(t)
' then A(t) • A + O
(3.9)
0
L:I
t
k=lO
[Sk(x(s»A(s) +A(s)Sk(x(s»
Io [B(x(s»A(s) +A(s)B(x(s»
..
* Jdek(s)
+ LJ Sk(x(s»A(s)Sk(X(S» k=l
where
and a(x) • crcr*(x)
* +a(x(s»Jds
419
Lecture iFl Some Preliminary Applications: In view of Theorem solution
x (t)
to
(3.7)
and Corollary (1.21)
admits a
(3.8)
,we know that the
N
co
C (R ) - density whenever we can b
get estimates of the form:
(4.1) where
b(t). det(A(t»
estimates like
(4.1)
A(t)
(3.9)
is given by
Obviously,
only can Come from an analysis of equation
(3.9)
To understand a linear fashion.
and
observe that
A(·)
(3.9)
enters this equation in
Hence one suspects that the time-honored method of
variation of parameters should be tried.
After a little thought, one
sees that: A(t) • X(t)AOX(t) * +
(4.2 ) where
X(t)
c
X(O,t)
and
Jt X(s,t)a(x(s»X(s,t) *ds o
o
X(s,t)
s
t
,is the solution to:
d
(4.3)
X(s,t)
1+
+
zstsk(X(u»X(S,u)dek(u) k=l s
St B(x(u»X(s,u)du s
t
s
For those who are not familiar with the variation of parameter technique in this context, the best way to check A(·)
(4.3)
is uniquely determined (path-wise) by
that the right hand side of As soon as one has classical result that definite.
(4.3)
(4.3) x(t)
Indeed, since
satisfies
is the first note that
(3.9)
and then verify
(3.9)
, i t is an easy matter to recapture the
has a smooth density if
(','J
a(·)
is positive
is a non-negative bilinear form,
420 A O
0
and so
I t X(s,t)a(x(s»X " (s,t)ds
A(t) Thus, if
EI
8 (.)
o
E> 0
,where A(t)
,then
t
EJ. X(s,t)X' (s,t)ds
:2'
o
Hence our problem comes down to estimating
o ,;;
Y(s,t) • X(s,t)X(s,t)"
(4.4 )
from below.
Perhaps the easiest way to obtain such estimates is to
derive an equation for (4.5 )
s ,;; t
X(s,t)
-1
X(s,' )
-1
t
.. I -
, namely: X(s,u)
-1
Sk(x(u»dBk(u)
k=l s
+ That (4.5)
X(s,' )
-1
St X(s,u) -1 [-B(x(u»+ s
sat i s f Les
(4.5)
and checking that
2
can be seen by defining
X(s,t)X(s,t)
-1
E
]du
k=l Xes,')
(4.6) t
Y(s,t)
+ ; [(Y(s,u) s where (4.7)
-1
-1
.. I -
*
r,S fY(s,u) kal s
,-B(x(u» +
d
t
k=l
(Sk(X(u»
** fCl,C2}.ClC2+C2Cl Tr(Y(s,t»
-1
Given -1
2
(4.5)
,Sk(x(u»
*
St s
where
*Y(s,u) -1 Sk(x(u»)du In particular,
d -1 2:; Yk(s,u)Tr(Y(s,u) )dBk(u)
kal s
+
,one finds that:
) 1+ ISk(x(u» kal
t
.. N - 2
to
*}dBk(u)
C , C2 E RN 0RN l
for
by
The argument involves
I
elementary Ito calculus plus the pathwise uniqueness of solutions linear stochastic integral equations.
-1
-1
)du
421
Tr(Y(s,u)
-1
Sk(x(u»
* )jTr(Y(s,u) -1 )
and
8(s,u) - Tr(Y(s,u)
-1
[-2B(x(u»
*+
:6d-, (2(Sk(x(u»
2
1,
) ; Sk(x(u»
*Sk(x(u»)J)jTr(Y(s,u) -1 )
k-1 Since
Y(s,u)
for any
is non-negative definite,
CERN 0 RN
Moreover,
Tr(Y(s,t)
-1
(4.7)
-1
,and therefore the
\Tr(Y(s,u) Yk's
-1-1 \\cllopTr(y(s,u) )
and the
e
are bounded.
is equivalent to:
d d ,-, t t \' 2 ) - Nexp[- 62SYk(s,u)dek(u)+S (B(s,u)+2 L;""k(s,u»du] k=l 5 s k-1
Hence, by standard estimates, for
1
p
0
so that
for
t(t) .. F(9(t) +"At)
Since
Os;ss;t it is now clear why the distribution of
s > 0
fails to have a density in
2 R
when
on an open interval containing Example (4.16):
0
lJ2 a' a +
dT\(t) .. CI('T\(t»d9(t)
max 19 (s) +}.s \ S;s and
I 3 Xl
one sees that
We can therefore find an
t" 0
E
In order to understand what is happening
,. .. inft t " 0 : '11 (t)
E [0,,.]
any
and
CI"i
throughout an open interval
0
here, firs t observe that
F E C"'(R
1/2
On the other hand, if
ill - probability.
i f there is a
+
and
0
ci
+ CI'e
-ae'
+ CI'e + lJ2 d'CI
2
for x(t)
vanishes
Xl
The preceding example deals with a situation to
which Hormander's theorem could have been applied.
We now want to
look at a situation which does not lend itself to analysis via Hormander's theorem. Let a =
(J:
-t
d d R ® R
and
Assume that for some
00*
where
d R
a(N)
determined by
«aij»ls;i,jsN (3.8)
d b: R
-t
d R
1 S; N
d
- functions.
and
s > 0
Define
a (N) (.) 0 all
428
xes) ,o(s) E X("')
that one should have to check is that His reasoning would be that 2 o (s)]ds
such optimism.
S [(o(s),o(s»)+
The problem is that in general all that one can say is
s (x(s),x(s»)(o(s),c(s»)
°2 (s)
1/2
o
la(')\
Hence if
2
s (0(-),0('»)
while
d8(s) +
I
t
0
e(s)ds
2
for some
6(');>0(')"£
it is possible to show that not only must
(4.18)
negative (a fact we already know) but also However, this is a long way from one to conclude is that for admits a density.
a.e.
(s) \ 1/2
8 (t)2
°
t:>
t(t)
ID '" E lSo 'KiD)
E LP 0
be non; 0
; all that it allows
the distribution of
Without going into more details, let it suffice to
say that equations of the form
I
and
then:
St
(4.18)
in which
(0(s),x(s»)d9 +
aCt) = (X(S),0(S»)j(X(S),x(S»)1/2
(o(s),o(s») +
From
o
s:> 0 •
t
Unfortunately, I have not been able to justify
LP (lr)
,I(x(s),o(s») \2
(x(t),x(t»)
and
I
for all
0 and that from this one ought to be able to obtain estimates on
111/ (x(t),x(t»)\\ that
(x(t),x(t»);
t
(4.18)
are far more delicate than ones
For example, if \ (t) I t 1/2 ; 2 \t(s) \ d8(s) + t and clearly
is replaced by
I
, then
is not integrable even though
0
; 0) = 0
for all
(t)
t :> 0
is still conceivable that a careful analysis of equations like
It
(4.18)
might give some useful insights into these questions, but as yet the final word is not in. In order to end on a less sour note, let uS conclude with an example to which Malliavin's method can be applied with complete success. Ci
:
I R
->
(0,,,,)
be a uniformly positive n
:I:
0
,for some
Let
- function and let '1\:> 0
Consider the process
429
t (.)
given by
(4.19)
..
Jt !X(I p (s - u) II
o
0
co>
where
p
(t) ..
L,
It is not hard to show that
o
(4.19)
uniquely
determines a progressively measurable and
then sup
(4.20)
where
AO'
411011 4 1
AT211p
and
1
Cb(R )
more, t(t) E 1'(,,)
for each
-p"II\
L ([O,T])
B(T) .. 8\\0\\2 t
0
eB(T)T
\\p'\\2
Ll([O,T])
thUll
Further-
we have a special case of
the general situation described in the preceding paragraph; only in this special case will we be able to get integrability estimates on for
t > 0
The technique which we will use is the following. define
a(N) (x) E RN+l
and
Given
N2 2
b (N) (x ) E RN+l
by if
i =
°
if
and consider the progressively measurable process given by
x
(N)
N+l
:[O,al)xe .... R
430
(4.21)
o
It is easy to show that if
(N) (t ) • N
where
p(N)(t).
(O'(t o
Nt'"
P (N) (s -
0 N )(.)
,then
(N) (u)du)d8 (s)
0
(4.20):
,and therefore by
o
_
distribution of
under
SUP\\ f(N)\l < '" N;;,2 t cn (R1 )
(N)
, then
III
denotes the
tends weakly to the
t
(t)
under
\Jl
'Thus if we can show that
where
f (N) t
E
and, in addition,
of
(N) (dx ) • feN) (x)dx t t
.... 0
In particular, if
for each. T "> 0
distribution
x6
a
Ill[ E sup
as
+ Stb(N)(X(N)(S»dS
X(N)(t). s.t a(N)(X(N)(S»d8(S)
,
then
=
f
t
(x j dx
where
f
t
E
b
Notice that for each situation treated in feN) t
E
estimates on that if
(4.16)
exists.
b
N., 2
Thus for each
defined aa in
(4.3)
E) .... 0
fast rate which is independent of
N., 2
\l.'(
sup
and
N;;, 2
The difficult ingredient here is to show
-1\"
then
t"> 0
What is not so clear is how to get uniform
\If(N)\\ t Cn(Rl)
X(N)(s,t)
,we have the "partially elliptic"
jX(N)(s,t)
00
suffice it to say that one has to X(N)(S,t) • I
as
and I) lOa t a lIufficient ly Without going into details,
use
+ St S(N)(X(S»X(N)(s,u)d8(u) s
+ stB(N)(X(S»X(N)(S,u)d u s and observe that estimates on made to depend on
III (
sup
Ix(N)(s,t)OO
V IlB(N)(X)\\op
-1\
E)
can be
alone (Le. independent
431
of
N
One then notes that if the
2 ).
with
introduced in connection
is strictly sl\\811er than one, then
can be bounded independent of 50
>. > 0
long as
>.
E (O,l)
distribution of
>.
If,
simply by replacing
N ':2:
with
teo)
2 I
V
\\B{N){X)\\op
This gives the required result ,one can reduce to the case
= t(t/2>.)
>. < I
and observing that the
is the same as the distribution of
defined
by ..
(y{IBp'(S o
0
and p(t) .. p (t/n) 2>. For more details on computations of the sort outlined above, see
where a{x) ..
1 (2).)1/2
section
in
(6)
(4)
432
References (1)
Horlllllnder, L., ''Hypoelliptic second order differential equations," Acta Math .• 119, pp. 147-171 (1967).
(2)
McKean, H.P., "Geometry of differential spaces," Ann. Pr ob , 1, pp. 197-206 (1973).
(3)
Stroock, D., and Varadhan, S.R.5 .• Multidimensional Diffusion Processes, Springer-Verlag (1979).
[4]
Stroock, D., "The Malliavin calculus and its applications to second order parabolic differential equations," Parts I and II, to appear in vol. 13 of Math. Systems Theory.
The probability functionals (Onsager-Machlup functions) of diffusion processes
Y. Takahashi, University of Tokyo and S. Watanabe, Kyoto University
Introduction. exp[ -1/2
Jo
For the n-dimensional Wiener measure, the functional
T
14>
t
I
2
dt]
is often considered as an ideal density with
respect to a fictitious uniform measure on the space of all continuous paths
Stratonovich [10]
introduced a notion of the
probability functionals of diffusion processes which may be considered as such ideal densities.
Also, physists call functions naturally associated
with these functionals the Onsager-Machlup functions and regard them as Lagrangeans giving rise to the most probable paths
[9],[2],[4],[7].
We are concerned with the following problem: given a non-singular, locally conservative diffusion process on a manifold
M, to obtain an
asymptotic evaluation of the probability that the paths of the diffusion lie in a small tube around a given smooth curve
$t: [O,T]-7 M.
Since
a Riemannian structure is naturally induced by the diffusion coefficients so that the generator of the diffusion is Beltrami operator,
l2
+ f
the Laplace-
f: a vector field) and an intrinsic metric defining
the tube should be the Riemannian distance
p(x,y), a precise formulation
of the problem may be given as follows: let
M be a Riemannian manifold
434
of the dimension n , I
2
generator
+ f
(x ,P) t
x
and
be the diffusin process with the
:[O,T]-+M
be a smooth curve.
Find an
asymptotic formula for the probability P", ( p (x
"o
e
t
t
)
0.
This convergence is clearly uniform in ••• ,K(j), ••• are independent of
Acknowledgment
r.
rE: 6(, since the constants
This
K 1,.
proof .
We would like to thank N. Ikeda and S. Kotani for
their valuable suggestions and discussions.
The above proof of Th.2.2
2 was suggested by N. Ikeda which simplified our original proof. L -conver-
gence in section 3 was suggested by Fujita and Kotani [3] where the same problem is discussed by a purely analytical method.
463 References [1]
E.Cartan; Lecons sur 1a geometrie des espaces de Riemann, GauthierVillars, Paris, 1963
[2]
D.Durr and A.Bach; The OnsagerMach1up function as Lagrangian for the most probable path of a diffusion process, Comm. Math. Phys. 60(1978) 153170.
[3]
T.Fujita and S.Kotani; The OnsagerMach1up functions for diffusion processes,
[4]
to appear.
R.Graham; Path integral formulation of general diffusion processes, Z. Physik B 26(1979), 281290
[5]
N.lkeda and S.Manabe; Integral of differential forms along the path of diffusion processes, Pub1. RIMS,Kyoto Univ. 15(1979), 827852.
[6]
N.lkeda and S.Watanabe; Stochastic differential equations and diffusion processes, KodanshaJohn Wiley, 1980.
[7]
H.lto; Probabilistic construction of Lagrangean of diffusion processes and its application,Prog.Theoretical Phys.
[8]
59(1978), 725741.
H.Kunita and S.Watanabe; On square integrable martingales, Nagoya Math.J. 30 (1967), 209245.
[9]
L.Onsager and S.Machlup; Fluctuations and irreversible processes, I, II, Phys. Rev. 91(1953), 15051512, 15121515.
[10]
R.L.Stratonovich; On the probability functional of diffusion processes, Select. Trans1. in Math. Stat. Prob. 10(1971), 273286.
ITO AND GIRSANOV FORMULAE FOR TWO PARAMETER PROCESSES. ATA AL-HUSSAINI UNIVERSITY OF ALBERTA, CANADA AND ROBERT J. ELLIOTI UNIVERSITY OF HULL, ENGLAND
Consider a single event that occurs at a random two parameter 'time'
T = (T1,T2)
2
E
• The underlying probability space can be taken to be Q
A probability measure given on
, which describes when the even occurs, is supposed
Q. In addition, we assume that
U (8 , 0 ) : 1
(8
1,82
)
2
E
=0
}
so that the event occurs on neither axis. The a-field F; , t be the product of the
a-fields F
on the two factors of
Q. F
{F
o
=v t
t
= a{L
t1 F
0
t
>
T1- 81
:
8
1
s t
1}
is defined to
E
= a{IT >
and F
t2
and F is the completion of F
0
2- 82
:
8
2 s t 2}
• The filtration
0
} considered is the right continuous completion of {Ft}.We suppose that T and T2 1
are independent under Write 1
F t1
x
x
2
Pt.
1.-
= I t :::.T.
Pt.
1.-
1.-
1.-
J
JO, t i"Ti J
and
(iu'"") dFiU. 1.-
1.-
i = 1,2.
E
Also, Pt. will be abreviated as Pi etc., and for suitable integrands 1 1.L
e,pi?2
t 2 = J0
etc.
465
DEFINITION A two parameter process X X
t ,t 1
ei
where
xt X
1
,t
t ,t 1
=X 2 =X 2
=X 2
i
E
0,0
0,0
+
e1 ,P
+
0,0
tr
U'P 1 + V.P
e2 ,P
e:5 ,P 1P2 'v
1P2 +
e4 ,P 1P2 rv
+
I"V
X can then be written either as
where
u =
where u
=
1,
+ u'P 2 +
is a SEMlMARTINGALE if it is of the form 2
rv
1P2 +
= 1,2,3,4.
t
1
e
e
'P 2 +
if, P1
+
.P
2
e
V
2,
J.p;, v
'P 2 +
3
e 'P 1
+
e
'P 2 or
4
e 'P 1
NOTATION If Y t
r
t
1 - Y is a process write 6 Y for the process Y t t ,t t i: t 2 1-,t2 1 2
2
2 6 y
t ,t 1
and 6Y
t 1,t2
for the process Y
t
+ Y 1,t2
t
for the process Y 2
- y 1-,t"2
t
- Y 1-,t2
t
1,t2
t
- Y 1,t2
t 1• t 2 -
-
Our form of Ito's formula is given by the following theorem. THEOREM Suppose X
t r t 2 is a semimartingale, as above, and
F:R
R is
a twice
differentiable function. Then F(X
t r t2)
= F(X
0,0
) + 6F(X)'P
1
P
2
PROOF.
Recall the differentiation formula for semimartingales in one dimensional time. There, if
f is a twice differentiable function and X a semimartingale
f (X ) = f(X ) + ftf,(X JdX +.lz t o e-: B
°
+
l ° f"(X
B-
Jd
B
466 We also quote the vector form of this result for the special case of a product:
where
[X,YJ
t
z
= (Xc,f)t +
Oee-ct:
tJX tS s s
In both time parameters the continuous part of our two paramter semimartingale X tl' t 2 is zero, so no predictable quadratic variation terms occur in our formulae. Holding
t
fixed and applying (]) to the X and F of our theorem:
2
+ I
[F(X
T
t
= F(X
0,0
)
+
J ] o
Similarly F'(X «.' = F'(X ) + u] 2 0,0
) tJX ] t ) - F(X - t ) - F' (X T] , 2 T],t T] -,t 2 2 t 6]F(X t )dp] + J ] F'(X _ t )V(u],t 2 0 ":: 2 ": , 2
I" 2
t
J 2
6
0
2F'(X )dP2 + u],u
2
A similar expression for F'(X Also F(X
t)
"i> 2
t ) = F(X
"i: 2
) +
0,0
/2
6
2F(X
0
with a similar formula for
V =
e2 .P 2
t
J 2 0
F"(X
u],u
jV(u],u
2
2
)d.P2•
(3)
(4)
is immediate.
)dP
2
+
/2 F' 0
(X
"r"»
jV(u],u
2
)d.P2
Using the product rule (2):
t )v(u],t 2) 2
F'(X _ u] ,
because
"i:":
)d.P r
+
e4 .P 2
Substituting for dv , and for dF' from (4), in this product formula, and finally substituting in (3) the desired identity is obtained.
GIRSANOV'S THEOREM Suppose to >L and write
is a new probability measure absolutely continuous with respect
467 for the martingale of conditional expectations. We also suppose that continuous with respect to
, so that
necessarily independent under L L
tl't2 t], t
> 0
a.s. Note that T] and T
is absolutely 2
are not
.
- ] is a centred martingale and so has a representation
2
where gEL] (Q) Therefore, L
t H(S],t 2)dQs] e t> 2
where H(t], t
2)
t
and H(tl't
2)
t
= L-]
-> t 2
fo
2
t]
= L-]
t l ' t 2-
fo
g(t],s2)dQ s2
g(s], t
2)dQs 2
Write
Analogues of Girsanov's formula are given by the following results:
THEOREM Consider a centred martingale X under measure
-.
t2
Xtt=fofo 2
r
is a weak martingale.
, so that
468 PROOF
We must show that L Write
t
r
t Nt 2
L
] + (LH).p] _. (LH).p]
N
(e.P2 = .p.p] -
et i
t
is a weak martingale under
•
2
+ H).P2)'P] -- (e(] + H).P2 +
e(R - (]
+ H) (] + H)).P2)·P]
'I'.p]
where
e,P
and
'I'
e(] + H),P
- e(] + H)'P
2
r
2
2
+ e(R - (1 + H)(1 + H))'P
2
Here, and in the sequel, the integrands are, where approproate, to be interpreted as left limits, so that, for example
(LH).p]
=
J
t]
_ t
L
os] ,
2
H(s],t
2)dP 1.
By the one-parameter product formula (2)
LN = (£.p).p] - (L'I').p] + (NLH).Ql + (LH).p]
(NLH).q] + L'I'.q] + (LK),P
1
where K = .p - 1jI + .pH.
However, using one-parameter calculus H
=
formula:
R
1 +
H
.p
2
- R'P ' and by the product 2
Substituting, we have that
+ e(] + H)(] +
-
=
R(e + .p)
] +
H
'P
However, we can also write
2
H) ·P2
- R(e +
0)
of the partition 1; of [o,tl goes to O. This decomposi-
tion into a martingale and a process "of zero energy" is unique.
477
Proof. 1) For a partition
(to, .. ,t
discrete Doob decomposition
(5)
M't"
t.
+
(=
0
of
[O,t]
A'l:
consider the
(i=O, .. ,n)
t.
1.
along
n)
1.
, where
for
i=O).
For d
>'l;'
we have
(6)
- 2
6
A _
since that
t
i+l
1.
ti
(A6 - A" ti t i+ l
)1
is a martingale along 17 . Thus, condition (1) implies 2. is a Cauchy sequence in L Define A - lim A17 t- \1:'loloO t ' as a right-continuous version of the square-integrable
martingale
Lt
L... t.
E [Xt - Atl Esl, At
i
/1 2
As = M - X s s 2( E[(M t-
and the right side converges to 0 as all paths of
A
Then
t
- At . ) 2] } ,
i
1.+1
1.
l't'l,j.. O. This implies that almost
are continuous, and so we have the existence of the
decomposition {3}. Its uniqueness as well as the extension from rO,t] to [O,oo}
is clear since a martingale with property (4) must be
constant. 2} Suppose that
X
is of the form (3). The argument in [4)p.91 shows tha t the random var iables in (5) converge to At in L 2 as 11:'1'" 0 . By (4), the last two sums in
converge to 0 . These two facts
imply via (6) that sup d,l.'l;"
converges to
0
as
0 , and this is condition (1).
Although a Dirichlet process is in general not a semimartingale, hence not an integrator for general predictable integrands by the theorem mentioned above, it does admit a pathwise ItO calculus in the
478 following manner. For a suitable sequence almost all trajectories of
X
i.e., there is an
have a quadratic variation along
process of the form
[X,xJ t
(7)
of partitions of [0,00),
+
=
such that, P-almost surely,
Ix.xl,
(8)
L
lim
=
n
in all continuity points of ties (7) and (8) and for any
So F ' (Xs- )dX s
n
l
[x,xl.
But for any trajectory with proper-
F EO C
t
lim n
2
, the stochastic integral
L
F ' (X ) (X - X t i t i + 1 t.) l
(;' .,.t . < t n
l
exists and satisfies the Ito formula
S
t
So F ' (Xs- ) dXs + L.[F(X )-F(X s
s-
+
)-F'(X
(O,t]
s-
)L:.x
FII(Xs_)d[X,XJs
s
- !F· t (X 2
s-
s
see [2J. References
[11
DELLACHERIE,C., et MEYER,P.A.: Probabilites et Potentiel, Ch.V-VIII Hermann, Paris (1980)
[2]
FOLLMER,H.: Calcul d 'lt6 sans probabilites. Preprint (1980)
[3]
FUKUSHlMA,M.: Dirichlet forms and Markov processes. North Holland (1980)
[4]
MEYER,P.A.: Integrales stochastiques. Sem.Prob.I, Lecture Notes in Mathematics 39, Springer (1967)
Brownian motion, negative curvature, and harmonic maps.
by W.S.Kendall, Department of Mathematical Statistics, The University, Hull.
This article discusses some relationships between the three topics mentioned in the title.
It is now well-known
that Brownian motion and complex function theory are closely linked, and that their interaction has bean very fruitful. A natural
generalisation of complex function theory is to 'harmonic'
functions taking their values in Riemannian manifolds. The report of Eells and Lemaire (1978) surveys work in this field. It is tempting to ask whether Brownian motion and probabilistic techniques have anything to offer; the article is an attempt to that they do. Details of proofs are not given; they will appear in a later production. It is hoped that their absence will allow the basic ideas to stand out clearly.
480
motion and the little Picard theorem.
1.
1.1
...............
Theorem (Picard) ... .... If
is holomorphic then it is constant.
f
Proof (after- Burgess Davis (1975) The complex Brownian motion
Z
BM(C)
is recurrent on open
sets e By L&vy's theorem (see
2.1 ) the image feZ) is a timeis nonconstant then the time-change is nontrivial. The idea of the proof then runs as follows. Let U be an open contractible neighborhood of the starting point of Z. If U is sufficientJy small, and the starting point is not at changed
BN( C \.:.1 ) • If
a degenerate point of
f
f
,then
feU)
will also be open and contractible.
There will be a peculiar random time T such that whenever t> T if Zt is in U then the path of feZ) up to time t will not be cor.tractible in C '\.:. 1 relative to feU) The proof that such a time T exists is the crux of the matter. Burgess Davis showed that such a
T
could be found, with
f(Z)\[O,t]
not null-homotopio in C\:!:1 when t> T and Zt E U • More recently McKean and Lyons (1980) have shown that 'not null-homotopic' can be replaced by 'not null-homologous', a stronger result correcting a contrary assertion of McKean (1969) •
Either way the existence of
a nonconstant holomorphic function recieves a topological-probabilistic contradiction, as can be seen from the diagram below.
c ';t
C
o
1
f
f(Z)/fo,t
zlfo,t) t
> T
1
481
I
(Levy) Holomorphic functions preserve
BM(C)
up to time changes.
There is a straightforward proof using the ItS calculus (see McKean(1969».
Naturally one asks oneself if such methods would work in higher dimensions, and for a first step one seeks a generalisation of
2.2
2.1.
(Fuglede(1978) , Bismut (1980), Ducourtioux, ••• ) Let
(M,g)
and
F: M
(N,h)
be riemannian manifolds. Then
N
is a harmonic morphism, preserving harmonic functions and thus Brownian motions up to time changes. if
(i)
dim M
dim N
(ii)
F
is harmonic ( in a generalised sensei see below);
(iii)
F
is horizontally conformal ( pee below).
Again one uses the
calculus.
The definitions involved appear at
2.3
and
Definition (see Eells and Lemaire (1978» is harmonic if
F
'i'J [ _ _ g . j
where the
r
+
o
are the Christoffel symbols of the manifolds.
It is illuminating to compare this with the formula for the differential generator of
BM(M,g)
• This generator is the Laplacian
482
operator
1::::..
for
M and is giTen by =
Mrijk
where both here and everywhere in the paper we are using the summation conTention for repeated indices. Note that the condition for
F
to be harmonic can be written
a
F
is semi conformal ( Fuglede terminology ) or horizontally
conformal ( Eells-Lemaire terminology ) if the on the derivative when
dF
a
dF
the restriction
dF \ ( Ker dF
is
conformal and surjective. As an exercise, check Fuglede's theorem 2.2 using these
,.
definitions and Ito's lemma 1
The fly in the ointment is best expressed by quoting the following result, now a classic; Theorem If
M=N
= Roo
and
m> 2
then a map that is conformal
in an open region is merely a restriction of a rigid motion that region. Since harmonic morphisms are essentially special conformal maps in such a context this result means THERE IS NO direct analogue
483
of the beautiful interplay between complex analysis and Brownian motion as soon as the dimensions get at all high 1 At least for Euclidean spaces of the same dimension the harmonic morphisms become very rare birds £
3.
Geometrical
eneralisations.
The holomorphic universal covering of Because
C
C
is
U
= ball
C •
is simply connected we can complete the dotted arrow
by a holomorphic
in the diagram below;
C-
--
a commuting diagram of holomorphic maps )
Once we have shown the existence of the covering map from
U to
C";!:,1
indicated in the diagram then the proof
of the little Picard theorem is merely an application of Liouville's theorem. However the existence of the covering map ( which must be holomorphic ) is related to elliptic function theory. Thus the Burgess Davis proof is a genuine alternative. This idea of a universal covering suggests a generalisation of the little Picard theorem. The surface
U can be given a metric
compatible with the analytic structure, and of constant negative curvature ( think of a saddle back to visualise negative curvature ). Skipping over the teChnicalities of curvature, we arrive at
484
(Goldberg, Ishihara, Petridis (1975) )
Theorem Let (a) also (b)
(M,g), (N,h)
be riemannian manifolds. Suppose
the Ricci curvature of
M is everywhere nonnegative-definite and
M is connected; the sectional curvatures of _H
0
N that
,there is no __ H and /8,T)one
has VA
('aG )dx x and G(xy) the kernel of (_lI+m2)1. For
=0
Remark: The proof for d
or Id = 2,
a.s.
= 2, Ia I < I41T
has been given in [1.26 b)]
(141T
is determined
by the fact that G(x) ;::; _ 1 In x as x :::0), The situation for /4; < co:;> /8,T is open. The "critical" value
I41T
is of course the same as the one for the
trigonometric interactions (SineGordon model) [I.27J. The proofs for
resp.
and d = 2,
resp. [1.26 d ) }. The case d = 2,
lal
>a
o
has been given in [1.26 c ) ]
lal > /8,T has been covered in [1.28].
509
Let now u(r) be any real-valued function onffi of one of the forms 2N s, 1) uCr-) asr a 0 2N>
I
2) u(r)
feardp(a)
f
3) u(r)
cos (ar + e)dp(a)
where p is any measure with support in (- 2 !;, 2 !;). In all cases we have n u(r) = L cnr for suitable c n' Define =: (X with A), n (X = A) n
L cn
:
: (X where A),
(X is such that A)
S) (X = U being the function in above theorem. We shall A A) an additive functional of the free field or shortly an
call
interaction (given by the function u). It is shown in all cases 1) - 3) (an small in 1), A small in 3)) that o
-
-UA
converges weakly as A
to a measure
on S'OR
2).
This
e
measure is Euclidean invariant (and yields relativistic quantum fields giving in particular the quantized solution of
For proofs of these statements see [1.29], [1.26], [1.27], (and references therein). The question now arises: can one put the constructed
in connection with
Dirichlet forms? This question was first discussed in [1.11 e)], [1.11 f)], [1.30] and following answer found. Let v be the restriction of generated by the fields
where
distribution u and QJ E
to the a-algebra
is the random fields with
where we use again the splitting ffid
=JR
x JRs, ffi
being the xO-axis. Then it can be proven [1.11 f)], [1.30] that for above models v has the property of the general theory of 1.2., so that again these v provide examples for the general theory.The relation between the diffusion process TIt associated with v and the random field
is such that the Osterwalder-Schrader
energy operator 6) H coincides as a form on twice-differentiable cylinder functions in L2(dv) with H • Next question: is
v
global Markov, so that one can, as in the free field case,
get directly Markov processes out of the associated random field?
510 7)
This is a difficult question
and it remained open for about 10 years.
By now one knows [I.30J that at least in the case where
is obtained
starting from a u of the trigonometric form 3), with A sufficiently small, then
is globally Markov. This settles the questions of existence of a non
Gaussian random field on 2
group
homogeneous with respect to the full Euclidean
and having the global Markov property.
9)
The precise theorem is the following Theorem
[1.30 a)J. Let d
e
=2
be the weak limit of
-U A
-
:u(.): (X A), A cos (ar+8),
U
- A e
A suf'f LcLenn.Ly small, 0 :;; 8
Then
and let