Concentrated Advanced Course January 7-8, 1999
and Workshop January 11-13, 1999 on
Pro duct Integrals and
Pathwise Integration
Foreword
In the week January 7-13, 1999 a course and a workshop on Product Integrals and Path-
wise Integration was held by MaPhySto at the Department of Mathematical Sciences,
University of Aarhus.
The course and workshop was organized by Ole E. Barndor -Nielsen Aarhus, Denmark ,
Svend Erik Graversen Aarhus, Denmark and Thomas Mikosch Groningen, The Nether-
lands .
In this lea et we have gathered the program, the list of participants and the workshop
abstracts. The notes for the course app eared as the rst volume of the MaPhySto Lecture
Notes series and may be fetched from our web-site www.maphysto.dk; hardcopies may
also b e ordered.
Contents
1 Course Program 2
2 Workshop Program 3
3 Combined list of participants to the Course and to the Workshop 5
4 Abstracts 12
1 Course Program
Thursday January 7
R. Dudley: How should integrals of Stieltjes type be de ned? .
9.30-10.30
Coffee/tea
R. Norvaisa: Stochastic integrals .
10.50-11.50
R. Dudley: Lyons' work on obstacles to pathwise It^o integration.
12.00-12.30
Lunch
12.30-13.30
R. Dudley: Di erentiability with respect to p-variation norms I .
13.30-14.15
R. Dudley: Di erentiability with respect to p-variation norms II .
14.20-15.05
Coffee/tea
R. Norvaisa: Properties of p-variation I .
15.20-16.00
R. Norvaisa: Properties of p-variation II .
16.15-17.00
Friday January 8
R. Norvaisa: Stochastic processes and p-variation.
9.30-10.30
Coffee/tea
R. Dudley: Empirical processes and p-variation.
10.45-11.45
Lunch
12.00-13.00
R. Norvaisa: Integration.
13.00-14.00
R. Dudley: Ordinary di erential equations and product integrals.
14.10-14.40
Coffee/tea
R. Norvaisa: Product integrals.
15.00-16.00
R. Dudley: Other aspects of product integrals.
16.10-16.40 2
2 Workshop Program
Monday January 11
Registration and coffee/tea
09.30-10.00
Chairman: Thomas Mikosch
Ole E. Barndor -Nielsen and Thomas Mikosch:
10.00-10.10
Introduction.
Richard Gill: Product-integration and its applications in survival
10.10-11.00
analysis.
Norb ert Hofmann: Optimal Pathwise Approximation of Stochas-
11.10-12.00
tic Di erential Equations.
Lunch
12.00-14.00
Chairman: Svend Erik Graversen
EskoValkeila: Some maximal inequalities for fractional Brownian
14.00-14.30
motions.
Richard Dudley: On Terry Lyons's work.
14.35-15.25
Coffee/tea
Rimas Norvaisa: p-variation and integration of sample functions
16.00-16.50
of stochastic processes.
S ren Asmussen: Martingales for re ected Markov additive pro-
17.00-17.50
cesses via stochastic integration.
Tuesday January 12
Chairman: J rgen Ho mann-J rgensen
Philipp e Carmona: Stochastic integration with respect to frac-
09.00-09.40
tional Brownian motion.
Rama Cont: Econometrics without probability: measuring the
09.45-10.35
pathwise regularity of price trajectories.
Coffee/tea
Francesco Russo: Calculus with respect to a nite quadratic
11.00-11.50
variation process.
Lunch
12.00-14.00 3
Chairman: S ren Asmussen
Pierre Vallois: Stochastic calculus related to general Gaussian
14.00-14.50
processes and normal martingales.
Imme van den Berg: Stochastic di erence equations, discrete
14.55-15.45
Fokker-Planck equation and nite path-integral solutions.
Coffee/tea
Rudolf Grub el: Di erentiability properties of some classical
16.10-17.00
stochastic models.
Donna M. Salop ek: When is the stop-loss start gain strategy
17.05-17.45
self- nancing?.
Discussion
17.50-18.30
Wednesday January 13
Chairman: Ole E. Barndor -Nielsen
Bo Markussen: Graphical representation of interacting particle
09.00-09.40
systems.
Jan Rosinski: Independenceof multiple stochastic integrals.
09.45-10.35
Coffee/tea
Zbigniew J. Jurek: In nite divisibility revisited.
11.00-11.50
Lunch
12.00-14.00 4
3 Combined list of participants to the Course and to
the Workshop
S ren Asmussen
Department of Mathematical Statistics
University of Lund
Box 118
S-221 00 Lund, Sweden
Email: [email protected]
Ole E. Barndor -Nielsen
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: o [email protected]
Jo chen Beisser
Chair of Banking
University of Mainz
Jakob Welder-Weg 9
D-55099 Mainz, Germany
Email: b [email protected]
Fred Esp en Benth
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: [email protected]
Preb en Blsild
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: preb [email protected]
Philipp e Carmona
Lab oratoire de Statistique et Probabilit es
Universit ePaul Sabatier
118 Route de Narb onne
31062 Toulouse Cedex, France 5
Email: [email protected]
Rama Cont
Centre de Mathematiques Appliquees
CNRS-Ecole Polytechnique
CMAP - Ecole Polytechnique
F-91128 Palaiseau, France
Email: Rama.Cont@p olytechnique.fr
Richard M. Dudley
MIT - Departmentof Mathematics
Ro om 2-245
Cambridge, MA 02139
U.S.A.
Email: [email protected]
S ren Fournais
Department of Mathematical Sciences
University of Aarhus
DK-8000 Aarhus C
Denmark
Email: [email protected]
Dario Gasbarra
Rolf Nevanlinna Institute
University of Helsinki PL 4
00014 Helsinki
Finland
Email: [email protected].
Richard D. Gill
University of Utrecht
Mathematical Institute
Budap estlaan 6
NL-3584 CD Utrecht, The Netherlands
Email: [email protected]
Svend Erik Graversen
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus, Denmark
Email: [email protected] 6
Rudolf Grub el
Universitat Hannover
FB Matematik
Welfengarten 1, PF 60 09
D-30617 Hannover, Germany
Email: rgrub el@sto chastik.uni-hannover.de
S ren Kold Hansen
Department of Mathematical Sciences
University of Aarhus
DK-8000 Aarhus C
Denmark
Email: [email protected]
Norb ert Hofman
Mathematisches Institut
Universitat Erlangen-Nurn b erg
Bismarckstr. 1 1/2
91054 Erlangen, Germany
Email: [email protected]
J rgen Ho mann-J rgensen
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: ho @imf.au.dk
Per Horfelt
Dept. of Mathematics
Chalmers University of Technology
Goteb org University
SE-412 96 Goteb org, Sweden
Email: p [email protected]
EvaVedel Jensen
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: [email protected]
Jens Ledet Jensen
Department of Mathematical Sciences 7
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: [email protected]
Wenjiang Jiang
Department of Mathematical Sciences
University of Aarhus
DK-8000 Aarhus C
Denmark
Email: [email protected]
Zbigniew J. Jurek
Institute of Mathematics
University of Wro claw
Pl. Grunwaldzki 2/4
50-384 Wro claw, Poland
Email: [email protected] c.pl
Marie Kratz
Universite Paris V
U.F.R. Maths/Info.
45 rue des Saints-Peres
75270 cedex 6Paris, France
Email: [email protected]
Martynas Manstavicius
Department of Mathematics
Vilnius University
Naugarduko
LT-2006 Vilnius, Lithuania
Email: [email protected]
Bo Markussen
Inst. for Matematiske Fag
K b enhavns Universitet
Universitetsparken 5
2100 K b enhavn , Danmark
Email: [email protected]
Thomas Mikosch
RUG Groningen
Faculty of Mathematics and Physics
PO Box 800 8
NL-9700 AV Groningen, The Netherlands
Email: [email protected]
Elisa Nicolato
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: [email protected]
Rimas Norvaisa
Institute of Mathematics and Informatics
Akademijos 4
2600 Vilnius
Lithuania
Email: [email protected]
Jan Parner
Biostatistisk Afdeling
Panum Instituttet
Blegdamsvej 3
1
b enhavn N, Denmark DK-2200 K
2
Email: [email protected]
Jan Pedersen
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: [email protected]
Jacob Krabb e Pedersen
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: krabb [email protected]
Goran Peskir
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email: [email protected] 9
Karsten Prause
Institut fur Mathematische Sto chastik
Universitt Freiburg
Eckerstrasse 1
D-79104 Freiburg, Germany
Email: [email protected]
Jan Ros nski
Department of Mathematics
121 Ayres Hall
University of Tennessee
Knoxville, TN 37996-1300, U.S.A.
Email: [email protected]
Francesco Russo
Universit eParis XIII
Institut Galil ee, Math ematiques
Av. Jean-Baptiste Cl ement
F-93430 Villetaneuse, France
Email: [email protected]
Donna M. Salop ek
Dept. of Mathematics
P.O.Box 800
University of Groningen
NL-9700 Av Groningen, The Netherlands
Email: [email protected]
Ken-iti Sato
Hachimanayama 1101-5-103
Tenpaku-ku
Nagoya 468-0074
Japan
Email:
Hansp eter Schmidli
Department of Mathematical Sciences
Theoretical Statistics
University of Aarhus
DK-8000 Aarhus C, Denmark
Email:
A. Reza Soltani 10
College of Sciences
Shiraz University
Shiraz
Iran
Email: [email protected]
Tommi Sottinen
Department of Mathematics
University of Helsinki
P.O. Box 4
FIN-00014
Email: tsottine@ro ck.helsinki.
Esko Valkeila
Compting Centre
University of Helsinki
P.O. Box 26
00014 Helsinki
Finland
Email: [email protected].
Pierre Vallois
Universit ede Nancy 1
D epartementde Math ematiques
B.P. 239
F-54506 Vanduvre les Nancy Cedex
France
Email: [email protected]
Imme van den Berg
Department of econometrics
University of Groningen
P.O.Box 800
9700 AV Groningen, The Netherlands
Email: [email protected]
Allan Wurtz
Dept. of Economics
University of Aarhus
DK-8000 Aarhus C
Denmark
Email: [email protected] 11
4 Abstracts
On the following pages you will nd the extended abstracts submitted to the organizers.
The contributions of R. Dudley and R. Norvaisa can b e seen as addendums to their lecture
notes An Introduction to p-variation and Young Integrals, MaPhySto Lecture Notes No. 1,
Aarhus, January 1999. 12
MARTINGALES FOR REFLECTED MARKOV
ADDITIVE PROCESSES VIA STOCHASTIC
INTEGRATION
S REN ASMUSSEN
Let X be an additive pro cess on a nite Markov pro cess J and
t t
consider Z = X + Y where Y is an adapted pro cess of nite variation,
t t t t
say the lo cal time at one or two b oundaries. We construct a family
of vector{valued martingales for J ;Z as certain sto chatic integrals
t t
related to the exp onential Wald martingales, thereby generalizing a
construction of Kella & Whitt 1992 for L evy pro cesses.
The applicability of the martingales is demonstrated via a number
of examples, including uid mo dels, a storage mo del and Markov{
mo dulated Brownian motion with two re ecting b oundaries.
Department of Mathematical Statistics, University of Lund, Box
118, S-221 00 Lund, Sweden
E-mail address : [email protected]
This is based on jointwork with O er Kella. 1
STOCHASTIC INTEGRATION WITH RESPECT TO
FRACTIONAL BROWNIAN MOTION
PHILIPPE CARMONA AND LAURE COUTIN
Abstract.For a general class of Gaussian pro cesses X , wich con-
tains typ e I I fractional Brownian motion of index H 2 0; 1 and
fractional Brownian motion of index H 2 1=2; 1 , we de ne a
sto chastic integral
Z Z
a s dX s = lim a s dX s
n
which is the limit of classical semi martingale integrals.
Extended Abstract
Fractional Brownian motion was originally de ned and studied by Kol-
mogorov within a Hilb ert space framework. Fractional Brownian
H
motion of Hurst index H 2 0; 1 is a centered Gaussian pro cess W
with covariance
1
2H
H H 2H 2H
E W W = t + s jt sj s; t 0
t s
2
1
for H = we obtain ordinary Brownian motion .
2
Fractional Brownian motion has stationary increments
2H
H H 2
E W t W s = jt sj s; t 0 ;
is H -self similar
1
d
H H
W ct ;t 0 = W t ;t 0 ;
H
c
and, for every 2 0;H , its sample paths are almost surely Holder
continuous with exp onent .
1
fractional Brownian motion is not However, in general i.e. H 6=
2
a semi-martingale see, e;g;, Decreusefond and Ustunel[1], Rogers ,
Salop ek . Therefore, integration with resp ect to fractional Brownian
motion cannot b e de ned in the standard way the semimartingale
approach .
Date : January 21, 1999.
1991 Mathematics Subject Classi cation. Primary 60G15, 60H07, 60H05 ; Sec-
ondary 60J65, 60F25 .
Key words and phrases. Gaussian pro cesses, Sto chastic Integrals, Malliavin Cal-
culus, Numerical Approximation. 1
2 P. CARMONA AND L. COUTIN
Existing de nitions of the integral. We are aware of at least three
ways of de ning the integral with resp ect to fractional Brownian mo-
tion. The rst metho d amounts to de ne a pathwise integral by taking
H
adavantage of the Holder continuity of sample paths of W . Given
a pro cess a such that almost all sample paths s ! a ! of a have
s
1
b ounded p variation on [0;t], for + H>1, the integral
p
Z
t
H
a s dW s
0
almost surely exists in the Riemann-Stieltjes sense see Young . Let
us recall that the p-variation of a function f over an interval [0;t]is
P
p
the least upp er b ound of sums jf x f x j over all partitions
i i 1
i
0=x 0 1 n The second metho d is to de ne the integral for deterministic pro cesses a 1 2 , then byan L isometry see Norros et al. . More precisely,ifH> 2 R H 2 a s dW s is de ned for functions a in L , the space of measurable functions f such that hhf; f ii < +1, with the inner pro duct de ned as Z Z 1 1 2H 2 hhf; gii = H 2H 1 f s g t js tj ds dt : 0 0 Eventually, the third metho d is the analysis of the Wiener space of the fractional Brownian motion see Decreusefond et Ustunel[1], and Duncan et al . On the one hand, the pathwise integral enables us to consider random 1 integrands. For instance, if H> , 2 Z t H H W s dW s 0 almost surely exists in the Riemann-Stieltjes sense. On the other hand, with the deterministic integral We require less regularity from the sample paths of the integrand a; we can compute the exp ectations " Z Z 2 1 1 H H E a s dW s =0; E a s dW s = hha; aii : 0 0 The class of integrators considered. The aim of this pap er is to de ne a sto chastic integral that tries to get the b est of b oth worlds, not only for fractional Brownian motion, but for a general class of Gaussian pro cess. The starting p oint of our approach is the construction of fractional Brownian motion given by Mandelbrot and van Ness [2]: Z t H 1=2 H 1=2 H dB ; W t = c u t u u H + + 1 3 where c is a normalizing constant, x = sup x; 0 denotes the p ositive H + part, and B ;u 2 R is a Brownian motion on the real line, that is u B ;u 0 and B ;u 0 are two indep endent standard Brownian u u motions. It is natural to consider the pro cess Z t V t = h t u dB h u 0 where h is lo cally square integrable. We shall consider more general pro cesses Z t W t = K t; u dB K u 0 where the kernel K is measurable and such that Z t 2 for all t> 0, K t; u du < +1 : 0 It is to b e noted that integrating on 0;t instead of 1;t causes no loss in generality. Indeed, see e.g. Theorem 5.2 of Norros et al , we can construct in this way fractional Brownian motion. More precisely, 1 for H> , W is fractional Brownian motion of index H for the choice K 2 Z t 1 1=2 H H 1=2 H 3=2 c s u u s du : K t; s = H H 2 s Another pro cess of interest in this class is typ e I I fractional Brownian motion of index H 2 0; 1 Z t 1 H 0 H 2 V t = c H t u dB t 0 u 0 2H 0 H 2 = t .It where c H is a normalizing constant such that E V t is a H self similar centered Gaussian pro cess whose almost all sample paths are Holder continuous of index , for 2 0;H ;however it has not stationary increments. The main idea. Let us explain now the main idea of this pap er. When h is regular enough, then V is a semi martingale. Therefore we h R can de ne a s dV s as an ordinary semimartingale integral, when a h is a nice adapted pro cess. In some cases we can restrict a to a space of good integrands and nd a sequence h converging to h in sucha way n R 2 that a s dV s converges in L to a random variable whichwe note h n R a s dV s . The space of go o d integrands will b e de ned by using h the analysis of the Wiener space of the driving Brownian motion B , and not the Wiener space of the pro cess V or W . h K Consequently, our rst result is to characterize the V which are semi- h p R of functions martingales. To this end weintro duce the space L + loc f which lo cally of p-th p ower integrable: 4 P. CARMONA AND L. COUTIN Z t p p f 2 L R if 8t> 0; jf s j ds < +1: + loc 0 0 From nowonh is a lo cally square integrable function such that h exists almost everywhere. 0 Theorem 1. The process V is a semimartingale if and only if h 2 h 2 L R . When this is the case, its decomposition is + loc Z t 0 v t =h 0 B + V s ds : h t h 0 We shall nowintro duce the space of go o d integrands the basic de ni- tons needed from Malliavin Calculus can b e found in Nualart's b o ok . Let I =[0;t] ; given q 2, we set for a 2 L I 1;2 Z Z Z 1=q 1=q t t t q q n a =E ja s j ds + E du dv jD a v j : q;t u 0 0 0 The space GI is the set of a 2 L such that n a < +1. q;t 1;2 q;t Theorem 2. Let p 2]1; 2[ and let q be the conjugated exponent of p p 1 1 0 + =1 . Assume that h 2 L R and a 2 GI . Then there + q;t loc p q R t exists a square integrable random variable denotedby a s dV s such h 0 that for every sequence h ;n 2 N of functions verifying n 0 2 1. for al l n, h and h arein L 0;t , n n 0 0 p 2. h 0 7 ! h 0 and h 7 ! h in L 0;t , n n n!1 n!1 2 we have the convergenceinL Z Z t t a s dV s 7 ! a s dV s ; h h n n!1 0 0 where on the left hand side we have classical semimartingales integrals. Let us stress the fact that Malliavin Calculus is a p owerful to ol that may totally disapp ear from the results, as show the following It^o's formula 2 R and that for Theorem 3 It^o's Formula . Assume that h 2 L + loc p 0 2 0 a p 2 0; 1 , h 2 L such that F and R . Then, for every F 2 C + loc 5 00 F are uniformly Lipschitz continuous, we have Z t 0 F V t = F 0 + F V s dV s h h h 0 Z Z t t 0 0 h u s F V u du dB = F 0 + h s s 0 Z Z t t 0 00 + h u s h u s F V u du ds h 0 s Z t 0 + h 0 F V s dB h s 0 Z t 1 2 00 + h 0 F V s ds : h 2 0 This Theorem needs the following comments observe rst that we can check this It^o's formula by computing the exp ectation of b oth sides Z t 1 00 2 E [ F V t ] = F V u h u du : h h 2 0 Furthermore, we can illustrate this formula by applying it to typ e 1 I I fractional Brownian motion of index H> : 2 Z Z t t H 2 H 3=2 H 2H V t = u s H 1=2 2V u du dB + t : s 0 s The next logical step of our study consists of showing that the integral we de ned coincides, for go o d regular integrands, with the pathwise 2 integrals. To this end, it is enough to show that it is the L -limit of Riemann sums over a re ning sequence of partitions of 0;t for then we can nd a subsequence converging almost surely . Theorem 4. Assume that p 2 0 R . h 2 L R and that for a p 2 0; 1 , h 2 L + + loc loc n ! is a sequenceofre ning partitions of [0;T] whose mesh n goes to 0. P n n a t 1 . = ! 0 where a a 2 GI is such that n a a i q;t q;t t ;t ] i i+1 3t n i 2 Then, we have the convergenceinL Z t X a t V t V t ! a s dV s : i h i+1 h i h 0 3t n i In order to cop e with true fractional Brownian motion, wehaveto establish the analog of Theorem 2 for pro cesses W .We assume that K for every t>0 6 P. CARMONA AND L. COUTIN Z 2 K t; s ds < +1 ; and that K t; s may b e written, for a measurable z , Z t K t; s = z u; s du 0 s s Eventually we let Z 1=p t p : jz s; u j ds kK k = sup p;t 0 u Theorem 5. Assume that for a p 2 0; 1 , we have kK k < +1 and p;t 1 1 that a 2 GI with + =1. q;t p q R t a s dW s Then there exists a square integrable random variable denotedby K 0 such that for every sequence K ;n 2 N of functions verifying n 1. for al l n, K has the representation n Z t K t; s = z u; s du 0 s n n s 2 with z s; : 2 L 0;s for every s n 2. kK K k ! 0. n p;t 2 we have the convergenceinL Z Z t a s dW s 7 ! a s dW s ; K K n n!1 0 where on the left hand side we have classical semimartingale integrals. 11 References 1. L. Decreusefond and A.S. Ustunel, Stochastic analysis of the fractional brownian motion,Potential Analysis 1997 . 2. B.B. Mandelbrot and Van Ness, Fractional brownian motions, fractional noises and applications, SIAM Review 10 1968 , no. 4, 422{437. 3. Nualart, David, The Mal liavin calculus and related topics., Probability and Its Applications, Springer-Verlag, New York, NY, 1995, [ISBN 0-387-94432-X]. Philippe Carmona, Laboratoire de Statistique et Probabilites, Uni- versitePaul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4 E-mail address : [email protected] Laure Coutin, Laboratoire de Statistique et Probabilites, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4 E-mail address : [email protected] ECONOMETRICS WITHOUT PROBABILITY: MEASURING THE PATHWISE REGULARITY OF PRICE TRAJECTORIES. RAMA CONT This talk deals with a pathwise approach to the analysis of prop erties of price variations. I use the notion of Holder regularity and p-variation to study sto ck price tra jectories from an empirical and theoretical p oint of view and compare empirical results with theoretical results on Holder regularity of familiar sto chastic pro cesses. Keywords: Holder regularity, singularity sp ectrum, wavelet trans- form, multifractal formalism, multiresolution analysis, Levy pro cess. Centre de Mathematiques Appliquees, CNRS-Ecole Polytechnique, CMAP - Ecole Polytechnique, F-91128 Palaiseau, France E-mail address : [email protected] 1 PICARD ITERATION AND p-VARIATION: THE WORK OF LYONS 1994 RICHARD M. DUDLEY REVISED 13 JANUARY 1999 - MULTIDIMENSIONAL CASE Abstract. Lyons 1994 showed that Picard iteration, a classical metho d in ordinary di erential equations so on to be recalled, extends to certain non-linear integral and di erential equations in terms of functions of b ounded p-variation for p < 2. The pro duct integral is a sp ecial case of Lyons's construction. 1. Classical Picard iteration. Supp ose given a non-linear ordinary di erential equation dy =dt = f y 1 1 where f is a C function from R into R with a uniformly b ounded derivative, so that f is globally Lipschitz. Supp ose we have an initial value y 0 = a and we are lo oking for a solution for t 0. There is a corresp onding integral equation Z t y t = a + f y u du; t 0: 2 0 Then y is a solution of the integral equation 2 if and only if it is a solution of 1 with y 0 = a. Consider the sequence of functions y de ned by y a and n 0 Z t y t := a + f y u du: 3 n+1 n 0 1 Then for n =0; 1; ; y is a C function on [0; 1 with y 0 = a. We would like to n n 0 nd conditions under which y converges to a solution. Let M := kf k := sup jf j. n L Then for n 1, Z t j y y t j M j y y u jdu: 4 n+1 n n n 1 0 R t MY u du for each t 0 and Let Y t := sup j y y u j: Then Y t n n n n 1 n+1 0ut 0 n 1 n n = 1; 2; . We have Y t tjf a j. Inductively, we get Y t jf a jM t =n! 1 n Mt for n = 1; 2; . Since the Taylor series of e converges absolutely for all t, we get that y t converges for all t 0 to some y t , uniformly on any b ounded interval n [0;T], where y is a solution of 2 . Moreover, the solution is unique: let y and t z be two solutions of 2 . Then j y z t j s M j y z u jdu: Let S t := 0 sup j y z u j. Then S t MtS t . Thus for t<1=M we get S t = 0. So we get 0ut y z successively on [0; 1= 2M ], [1= 2M ; 1=M ], [1=M; 3= 2M ]; ; so y t = z t for all t 0. Alternatively, convergence of fy g could be proved as follows: by 4 , n Y t MtY t , so restricting to 0 t 1= 2M , absolute and uniform convergence n+1 n 1 2 RICHARD M. DUDLEY follows from a geometric series, and again one can iterate to further intervals of length 1= 2M . 2 Example. Consider the analytic but not globally Lipschitz function f y = y . Solving 2 1 by separation of variables gives dy =y = dt, 1=y = t + c, y = 1= t + c . For a>0, setting c := 1=a we get a solution y of 1 and 2 for 0 t<1=a which go es to 1 as t " 1=a. 2. Lyons's extension and composition. d Let S; jj b e a normed vector space. The two main examples in view will b e S = R with its usual Euclidean norm j j and S = M , the space of d d real matrices A, with d 0 d 0 the matrix norm kAk := sup fjAx j : x 2 R ; jxj =1g where x isa1 d vector and x its transp ose, a d 1 column vector. For a function f from an interval [0;T]into S and 0 p d norm on S . Thus if S = R , f = f ; ;f with f : [0;T] 7! R and v f v f for 1 d j p j p each j . The set of functions f : [0;T] 7! S with v f < 1 will b e called W [0;T];S , p p R T d or just W if S = R. If H and x map an interval [0;T]into R then H dx will b e p 0 R P T d de ned as H t dx t ifthed integrals exist. If instead H maps [0;T]into M j j d j =1 0 R T d and again x maps [0;T]into R then H dx will b e the vector whose ith comp onent 0 R P T d 2 is H t dx t if the d integrals exist. ij j j =1 0 Lyons 1994 considered an extension of 2 with Z t z t = a + F z u dx u ; 0 t T; 5 0 d d d where F : R 7! M is a suitable function, a 2 R , and x 2W [0;T]; R for some p, d p 1 p<2. In Lyons's work, since the functions app earing are continuous, the integrals can be taken in the ordinary Riemann-Stieltjes sense. Note that the integral equations for the pro duct integral CAC Sections 7.1, 7.4 have the same form as 5 where F is the identity function except that here z and x are vector-valued rather than matrix- valued. Lyons discovered, and it will b e proved b elow, that the equation can be solved by Picard iteration whenever F has a gradient OF everywhere, satisfying a global Holder condition of order where p < 1+ 2. The integral will exist by the Love-Young d 1 1 inequality CAC, section 4.4 if F z 2W [0;T]; R with p + q > 1. Moreover, we q have: d Theorem 1. On an interval [0;T], if f 2 W [0;T];M and g 2 W [0;T]; R with q d p R x 1 1 d + q > 1, then for h x := , p f dg , the inde nite integral h is in W [0;T]; R p 0 1 1 with khk p + q kf k kg k : p [q ] p Pro of. This follows from the pro ofs in Section 4.4 of CAC, which don't require that f and g have scalar values. The integrals are in the re nement-Young-Stieltjes sense, see CAC sections 2.1, 2.2 and 6.2. p Previously,we had considered comp osition op erators f; g 7! F + f G + g 2L r with f in spaces W and G; g 2 L , 1 p < r , and suitable G CAC, Section 3.3 . p 1 Note that for a C function G we can have F G non-regulated for F 2W W for 1 p PICARD ITERATION AND p-VARIATION: THE WORK OF LYONS 1994 3 1=x all p, sp eci cally F := 1 , G x := e sin 1=x , x 6= 0, G 0 := 0. Here we [0;1 d d will consider the Nemitskii op erator g 7! F G + g : W [0;T]; R 7! W [0;T]; R r r for xed suciently smo oth F , as follows. Let S; j j and U; kk be normed vector spaces. For a subset C S and 0 < 1, H C; U is the space of Holder functions f : C 7! U with seminorm kf k := sup kf x f y k=jx y j < 1; f g x6=y and H denotes the space of b ounded functions in H with the norm ;1 kf k := kf k + kf k sup f g f g where the supremum norm is denoted by khk := sup jh x j and the supremum is sup x over the domain of h for any b ounded function h. Let H := H R; R . Recall that on a b ounded interval in R, H W . To see that H is not included in W for any 1= p p<1= one can consider the lacunary Fourier series examples of L. C. Young, e.g. CAC, pro of of Theorem 4.29. For function spaces F ; G ; H , F G H will mean that f g 2 H for all f 2F and g 2G. The following are then easy to verify: For any three normed spaces S; U; V , H U; V H S; U H S; V for 0 < ; 1; H H is not H if > for 0 < ; 1: consider g x := x , f y := y , 0 x; y 1. For normed spaces S; U and an interval [0;T], H S; U W [0;T];S W [0;T];U , p p= for 0 < 1, 1 p < 1, f 2 H S; U , with kf g k kf k kg k p= f g p g 2W [0;T];S . p It is known, moreover, that for S = U = R, given 0 < 1 and 1 p<1, f 2 H is not only sucient, but necessary so that ff g W W : see Ciemno czolowski and p p= Orlicz 1986 , Theorem 1, in which we supp ose that \ 1 for x" means \< 1 for all x ." 1 d For 0 < 1 the space H will b e de ned as the set of C functions : R 7! R 1+ ;d d d d d such that O 2 H R ; R . Also, H will b e the set of functions F : R 7! M for ;1 d 1+ ;d which each entry F 2 H . The next theorem gives a kind of Lipschitz prop erty of ij 1+ ;d d the Nemitskii op erator g 7! G + g : W [0;T]; R 7! W [0;T]; R under sucient r r assumptions on ;G. The theorem is not in Lyons's pap er and seems to provide a shorter pro of than his, not requiring his new extension of the Love-Young inequality cf. CAC, section 7.5 but only the original inequality. We have: d d Theorem 2. If 0 < 1, 1 r < 1, g 2 W [0;T]; R , and G 2 W [0;T]; R , r r with 2 H ; we have G + g and G 2W [0;T]; R , k G + g Gk 1+ ;d r sup kO k kg k , and sup sup k G + g Gk kO k kg k + kg k kO k kGk : r sup r sup f g r d Pro of. Wehave G 2W [0;T]; R and since 2 H we have G + g and G in r 1 W [0;T]; R as noted ab ove. The second conclusion is also clear. For a t 4 RICHARD M. DUDLEY have j G + g u G u G + g t + G t j S + T where S := j G + g u G u + g t j kO k jg u g t j; sup T := j G u + g t G + g t G u + G t j: If g t = 0 then T =0. Otherwise g t =jg t j is a unit vector and " ! ! Z jg t j g t g t g t T = O G u + s O G t +s ds jg t j jg t j jg t j 0 jg t jkO k jG u G t j : f g For any partition 0 = x 0 1 n S and T to t = x and u = x , j = 1; ;n, sum the r th powers over j and apply j 1 j Minkowski's inequality, proving Theorem 2. The next theorem shows how Picard iteration works under Lyons's conditions. Denote d d the set of continuous functions in W [0;T]; R by C W [0;T]; R . p p d Theorem 3. Lyons Let 0 < 1, 1 p < 1+ and x 2 C W [0;T]; R p d d for some T > 0. Let F 2 H . For a 2 R let z t a, and for n = 1; 2; ; 0 1+ ;d R t d F z u dx u . Then z converges in W [0;T]; R to some z which z t := a + n 1 n p n 0 is the unique solution of 5 for 0 t T . Pro of. Theorem 2 will b e applied with r := p= and where = F for each i; j . Then ij 1 1 1 d d r + p = p 1 + > 1. Wehave W [0;T]; R W [0;T]; R since p p r d d z 2 W [0;T]; R . It will be shown inductively that z 2 W [0;T]; R for each n. 0 p n p d Supp ose z 2 W [0;T]; R . Then F z 2 W [0;T];M W [0;T];M since n 1 p n 1 p d r d d F is Lipschitz. Thus by Theorem 1, z 2W [0;T]; R as stated. n p Let khk be the p-variation seminorm of h on [0;t]. For each n =0; 1; 2; and [0;t]; p t > 0, let A t := kz z k . Note that z z 0 = 0 for all n. Thus n n+1 n [0;t]; p n n 1 kz z k kz z k for 1 p 1. Then Theorem 2 with g := z z n n 1 sup n n 1 n n 1 p and G := z gives by the Love-Young inequality in the form of Theorem 1 n 1 A t B 2C + D kGk N A t 6 n t n 1 p P P d d kOF k , and kOF k , D := where B := 1 + =p ; C := ij ij sup f g i;j =1 i;j =1 N := kxk . We have t [0;t]; p A t = kz k jF a jN 1 7 0 1 [0;t]; p t for 0 t small enough. Take 0 < small enough so that 1 1 B 2C +2D N < 1=2 8 t for 0 t . It will b e shown by induction on n that for 0 b oth n 1 A t A t =2 2 ; 9 n+1 n and 1 n kz k 2 2 : 10 n [0;t]; p PICARD ITERATION AND p-VARIATION: THE WORK OF LYONS 1994 5 For n = 0, 10 holds, as do es the right inequality in 9 by 7 , which also gives 10 for n = 1. For each n = 0; 1; 2; ; 10 and 8 for n, and 6 for n +1, imply the rst inequality in 9 . Thus, inductively, the second inequality also holds. Since kz k kz k + A , 10 and 9 for a given n imply 10 and thus 9 for n +1, n+1 n n completing the induction step and the pro of of 9 and 10 for all n. From 9 it follows that fz g is a Cauchy sequence for kk , so it converges to some z 2W [0;]. n [0;t]; p p 0 0 We can rep eat the whole pro cess on an interval [; + ] for some > 0. Since kxk < 1, and B; C; D remain constant, after nitely many steps we have z [0;T ]; p de ned on the whole interval [0;T]. By continuity, z satis es 2 . For uniqueness, supp ose y is another solution of 5 on [0;T]. Let B t := ky z k . Then by the pro of of 9 , B t B t =2 for t ; so B t =0 on[0;] and by [0;t]; p iteration, y t =z t on [0;T]. Theorem 3isproved. REFERENCES CAC = R. M. Dudley and R. Norvaisa, An Introduction to p-variation and Young Integrals. Maphysto, Aarhus, January 1999. J. Ciemno czolowski and W. Orlicz, Comp osing functions of b ounded -variation. Proc. Amer. Math. Soc. 96 1986 , 431-436. T. Lyons, Di erential equations driven by rough signals I : An extension of an in- equalityofL. C. Young. Math. Research Letters 1 1994 , 451-464. APPLICATIONS OF PRODUCT-INTEGRATION IN SURVIVAL ANALYSIS RICHARD GILL In the talk I will describ e a number of applications of pro duct- integration in survival analysis. In particular I will fo cus on the Kaplan-Meier estimator use of the Duhamel equation to study its prop erties the Aalen-Johansen estimator of the transition probability ma- trix of a markov chain based on observations with censoring and delayed-entry the Dabrowska multivariate pro duct-limit estimator. In each case I will use di erent probabilistic metho ds, namely: function- indexed empirical pro cesses and the van der Laan identity; martingale metho ds; and compact-di erentiability metho ds resp ectively. Please nd app ended my contribution to the Encyclopaedia of Sta- tistical Science, which serves as a brief intro duction and contains useful references. Mathematical Institute, University of Utrecht, The Netherlands E-mail address : [email protected] 1 Pro duct-Integration ESS P Q All statisticians are familiar with the sum and pro duct symb ols and , and with the integral symbol R . Also they are aware that there is a certain analogy b etween summation and integration; in fact the integral symb ol is nothing else than a stretched-out capital S |the S of summation. Strange therefore that not many p eople are aware of the existence of the pro duct-integral ,invented by the Italian mathematician Vito Volterra in 1887, which b ears exactly the same relation to the ordinary pro duct as the integral do es to summation. The mathematical theory of pro duct-integration is not terribly dicult and not terribly deep, whichis p erhaps one of the reasons it was out of fashion again by the time survival analysis* came into b eing in the fties. However it is terribly useful and it is a pity that E.L. Kaplan and P. Meier [10], the inventors 1956 of the pro duct-limit or Kaplan-Meier* estimator the nonparametric maximum likeliho o d estimator of an unknown distribution function based on a sample of censored survival times , did not make the connection, as neither did the authors of the classic 1967 and 1974 pap ers on this estimator by B. Efron [5] and N. Breslow and J. Crowley [3]. Only with the 1978 pap er of O.O. Aalen and S. Johansen [1] was the connection b etween the Kaplan-Meier estimator and pro duct-integration made explicit. It to ok several more years b efore the connection was put to use to derive new large sample prop erties of the Kaplan-Meier estimator e.g., the asymptotic normality of the Kaplan-Meier mean with the 1983 pap er of R.D. Gill [8]. Mo dern treatments of the theory of pro duct-integration with a view toward statistical applications can b e found in [2], [6], and [9]. The Kaplan-Meier estimator is the pro duct-integral of the Nelson-Aalen estimator* see counting pro- cesses* of the cumulativeorintegrated hazard function*; these two estimators b ear the same relation to one another as the actual survival function* one minus the distribution function and the actual cumulative hazard function. There are many other applications of pro duct-integration in statistics, for instance in the study of multi-state pro cesses connected to the theory of Markov pro cesses as initiated by Aalen and Jo- hansen [1] and in the theory of partial likeliho o d cf. Cox regression mo del* ; see Andersen, Borgan, Gill and Keiding [2]. Pro duct-integration also turns up in extreme-value theory where again the hazard rate plays an imp ortant role, and in sto chastic analysis and martingale* theory sto chastic integration , where it turns up under the name Dol eans-Dades exp onential martingale. Prop erties of integrals are often easily guessed by thinking of them as sums of many, many usually very small terms. Similarly, pro duct-integration generalises the taking of pro ducts. This makes prop erties of pro duct-integrals easy to guess and to understand. Let us de ne pro duct-integration at a just slightly higher level of generality than Volterra's original de nition corresp onding to the transition from Leb esgue to Leb esgue-Stieltjes integration . Supp ose X t is a p p matrix-valued function of time t 2 [0; 1 . Supp ose also X or if you prefer, each comp onentofX is right continuous with left hand limits. Let 1 denote the identity matrix. The pro duct-integral of X over the interval 0;t]isnow de ned as t Y 1 +dX s = lim 1 + X t X t i i 1 max jt t j!0 i i 1 0 where the limit is taken over a sequence of ever ner partitions 0 = t 0 1 k interval [0;t]. For the limit to exist, X has to b e of b ounded variation; equivalently, each comp onentofX is the di erence of two increasing functions. Avery obvious prop erty of pro duct-integration is its multiplicativity. De ning the pro duct-integral over an arbitrary time interval in the natural way,wehave for 0 t s t 1 +dX = 1 +dX 1 +dX : 0 0 s We can guess for pro ofs, see [6] or preferably [9] many other useful prop erties of pro duct-integrals by lo oking at various simple identities for nite pro ducts. For instance, in deriving asymptotic statistical theory it is 1 often imp ortant to study the di erence b etween two pro duct-integrals. Nowifa ,:::,a and b ,:::,b are 1 k 1 k two sequences of numb ers, wehave the identity: Y Y X Y Y 1 + a 1 + b = 1 + a a b 1 + b : i i i j j i j i This can b e easily proved by replacing the middle term on the right hand side of the equation, a b , by j j 1 + a 1 + b . Expanding ab out this di erence, the right hand side b ecomes j j X Y Y Y Y 1 + a 1 + b 1 + a 1 + b : i i i i j i>j i>j 1 ij ij 1 This is a telescoping sum; writing out the terms one by one the whole expression collapses to the two outside pro ducts, giving the left hand side of the identity. The same manipulations work for matrices. In general it is therefore no surprise, replacing sums byintegrals and pro ducts by pro duct-integrals, that t t t s Z t 1 +dY : 1 +dX 1 +dY = 1 +dX dX s dY s s=0 s+ 0 0 0 This valuable identity is called the Duhamel equation the name refers to a classical identity for the derivative with resp ect to a parameter of the solution of a di erential equation . b As an example, consider the scalar case p = 1 , let A b e a cumulative hazard function and A the Nelson-Aalen estimator based on a sample of censored survival times. In more detail, we are considering the statistical problem of estimating the survival curve S t =PrfT tg given a sample of indep endently censored* i.i.d. survival times T , :::, T . The cumulative hazard rate A t is de ned by 1 n Z t dS t A t = ; S t 0 A is just the integrated hazard rate in the absoutely continuous case, the cumulative sum of discrete hazards in the discrete case. Let t 1 2 j numb er of individuals at risk just b efore time t and let d denote the numb er of observed deaths at time j j t .We estimate the cumulative hazard function A corresp onding to S with the Nelson-Aalen estimator j X d j b A t = : r j t t j This is a discrete cumulative hazard function, corresp onding to the discrete estimated hazard b t =d =r , j j j b b t = 0 for t not an observed death time. The pro duct-integral of A is then t Y d j b b S t = ; 1 dA = 1 r j t t j 0 which is nothing else than the Kaplan-Meier estimator of the true survival function S . The Duhamel equation now b ecomes the identity Z t S t b b b S s dA s dA s S t S t = S s s=0 which can b e exploited to get b oth small sample and asymptotic results, see Gill [7,8,9], Gill and Johansen [6], Andersen, Borgan, Gill and Keiding [2]. The same identitypays o in studying Dabrowska's [4] multivariate pro duct-limit estimator see [9], [11] , and in studying Aalen and Johansen's [1] estimator of the transition 2 matrix of an inhomogeneous Markovchain see [2] . It can b e rewritten [9] as a so-called van der Laan b [11] identity expressing S S as a function-indexed empirical pro cess, evaluated at a random argument, so that the classical large sample results for Kaplan-Meier consistency, asymptotic normality can b e got byatwo-line pro of: without further calculations simply invoke the mo dern forms of the Glivenko-Cantelli* theorem and the Donsker* theorem; i.e., the functional versions of the classical law of large numb ers and the central limit theorem resp ectively. Taking Y identically equal to zero in the Duhamel equation yields the formula s t Z t 1 +dX 1 = 1 +dX dX s : s=0 0 0 This is the integral version of Kolmogorov's forward di erential equation from the theory of Markov pro cesses, and it is the typ e of equation: Z t Y s dX s Y t = 1 + 0 t in unknown Y , given X , which motivated Volterra to invent pro duct-integration. Y t = 1 +dX is 0 the unique solution of this equation. References [1] O.O. Aalen and S. Johansen, 1978 , An empirical transition matrix for nonhomogenous Markovchains based on censored observations, Scandinavian Journal of Statistics 5, 141{150. Intro duced simultaneously counting pro cess theory and pro duct-integration to the study of nonpara- metric estimation for Markov pro cesses; the relevance to the Kaplan-Meier estimator was noted by the authors but not noticed by the world! [2] P.K. Andersen, . Borgan, R.D. Gill and N. Keiding 1993 , Statistical Models Based on Counting Processes, Springer-Verlag, New York 778 pp. . Contains a `users' guide' to pro duct-integration in the context of counting pro cesses and generalised survival analysis. [3] N. Breslow and J. Crowley 1974 , A large sample study of the life table and pro duct limit estimates under random censorship, Annals of Statistics 2, 437{453. First rigourous large-sample results for Kaplan-Meier using the then recently develop ed Billingsley-style theory of weak convergence. [4] Dabrowska, D. 1978 , Kaplan-Meier estimate on the plane, Annals of Statistics 16, 1475{1489. Beautiful generalization of the pro duct-limit characterization of the Kaplan-Meier estimator to higher dimensions. Other characterizations, e.g., nonparametric maximum likeliho o d, lead to other estimators; see [9], [11]. [5] Efron, B. 1967 , The two sample problem with censored data, pp. 831{853 in: L. LeCam and J. Neyman eds. , Proc. 5th Berkeley Symp. Math. Strat. Prob., Univ. Calif. Press. This classic intro duced the redistribute-to-the-right and self-consistency prop erties of the Kaplan-Meier estimator and claimed but did not proveweak convergence of the Kaplan-Meier estimator on the whole line in order to establish asymptotic normality of a new Wilcoxon generalization, results nally estab- lished in [7]. [6] Gill, R.D., and Johansen, S. 1990 , A survey of pro duct-integration with a view towards application in survival analysis, Annals of Statistics 18, 1501{1555. The basic theory, some history, and miscellaneous applications. [9] contains some improvements and further applications. 3 [7] Gill, R.D. 1980 , Censoring and Stochastic Integrals,MCTract 124, Centre for Mathematics and Com- puter Science CWI , Amsterdam. The author emphasized the counting pro cess approach to survival analysis, using some pro duct-limit theory from [1] but not highlighting this part of the theory. [8] Gill, R.D. 1983 , Large sample b ehaviour of the pro duct limit estimator on the whole line, Ann. Statist. 11, 44{58. Cf. comments to [7]. [9] Gill, R.D. 1994 , Lectures on survival analysis. In Lectures on Probability Theory Ecole d'Et ede Probabilit es de Saint Flour XXII - 1992 , D. Bakry, R.D. Gill and S.A. Molchanov, ed. P. Bernard, Springer-Verlag SLNM 1581 , Berlin, pp. 115{241. Perhaps cryptically brief in parts, but a yet more p olished treatment of pro duct-integration and its applications in survival analysis. [10] Kaplan, E.L., and Meier, P. 1958 , Nonparametric estimation from incomplete observations, J. Amer. Statist. Assoc. 53, 457{481, 562{563. This classic is actually numb er 2 in the list of most cited ever pap ers in mathematics, statistics and computer science fourth place is held byCox, 1972; rst place by Duncan, 1955, on multiple range tests . The authors never met but submitted simultaneously their indep endentinventions of the pro duct-limit estimator to J. Amer. Statist. Assoc.; the resulting joint pap er was the pro duct of p ostal collab oration. [11] van der Laan 1996 , Ecient and Inecient Estimation in Semiparametric Models, CWI Tract 114, Centre for Mathematics and Computer Science, Amsterdam. Contains a b eautiful identity for the nonparametric maximum likeliho o d estimator in a missing data problem: estimator minus estimand equals the empirical pro cess of the optimal in uence curveevaluated P n 1 b b at the estimator, F F = IC X ; F ; applications to the bivariate censored data problem as opt i i=1 n well as treatment of other estimators for the same problem. RICHARD GILL 4 DIFFERENTIABILITY PROPERTIES OF SOME CLASSICAL STOCHASTIC MODELS RUDOLF GRUBEL In statistics it is well known that di erentiability prop erties of an estimator can b e used to obtain asymptotic normalityortoinvestigate the validity of b o otstrap con dence regions. Here we concentrate on the use of di erentiability in connection with p erturbation asp ects and numerical treatment of sto chastic mo dels. Our rst example deals with the expansion of G/G/1 queues ab out M/M/1 typ e mo dels, in the second we discuss the applicability of Richardson extrap olation for the computation of p erp etuities. In b oth cases a decomp osition of the functional in question into simpler comp onents has to be achieved. In the rst example the general theory of commutative Banach algebras turns out to b e very useful, the analysis in the second example is based on the investigation of xed p oint equations. Universitat Hannover, FB Matematik, Welfengarten 1, PF 60 09, D-30617 Hannover, Germany E-mail address : [email protected] 1 Optimal Pathwise Approximation of Sto chastic Di erential Equations Universitat Erlangen-Nurnb erg Norb ert Hofmann Thomas Muller-Gronbach Freie Universitat Berlin Klaus Ritter Universitat Passau December 21, 1998 1 Global Error in L -norm 2 We study the pathwise strong approximation of scalar sto chastic di erential equations with additive noise dX t =a t; X t dt + t dW t ; t 2 T; 1 driven by a one-dimensional Brownian motion W ,on the unit interval T =[0; 1]. Di erent notions of errors for pathwise approximation are studied in the literature on Sto chastic Numerics. See Klo eden and Platen 1995 , Milstein 1995 , and Talay 1995 for results and references. Mainly errors of pathwise approximations are de ned discretely at a nite number of p oints in T . In most of the cases these p oints coincide with the discretization of the given metho d. We follow a new approach by measuring X globally the pathwise distance b etween the strong solution X and its approximation on T in the L -norm kk and de ning the overall error of X by 2 2 1=2 2 X = E kX X k : e 2 We aim at determining asymptotically optimal approximation metho ds. To that end we consider arbitrary metho ds X , that use observations of the Brownian motion W n at n p oints. Moreover, a nite number of function values or derivative values of a and may b e used. We establish sharp lower and upp er b ounds for the minimal error 1 e inf X that can be obtained by metho ds of the ab ove typ e. Note that upp er n X n b ounds may be shown by the error analysis of a sp eci c metho d, while lower b ounds must hold for every metho d X . The optimal order is achieved by an Euler scheme n with adaptive step-size control. We state this more precisely in the sequel. 2 Euler Scheme with Adaptive Step-Size Control For an arbitrary discretization 0= < < =1 2 0 n b of the unit interval the Euler scheme X , applied to equation 1 , with initial value X 0 is de ned by b X =X 0 0 and b b b X =X +a ; X + W W k +1 k k k k +1 k k k +1 k b where k =0;:::;n 1. The global approximation X for X on T is de ned by piecewise b linear interp olation of the data ; X with k =0;:::;n. Let k k h>0 be a basic step-size which we cho ose previously. We intro duce the following adaptive h b b step-size control for the Euler metho d X = X : Put = 0 and 0 2=3 = + min h ; h= ; 3 k +1 k k as long as the right-hand side do es not exceed one. Otherwise put =1. k +1 h b We study the asymptotic b ehaviour of the error e X with h tending to zero, where X t is the unique strong solution of an equation 1 . The error analysis based on the following assumptions on the drift a : T R ! R and the di usion co ecient : T ! R 2 A There exist constants K ;K ;K > 0 such that 1 2 3 0;1 0;2 a t; x K ; a t; x K ; 1 2 and ja t; x a s; x jK 1 + jxj jt sj 3 for all s; t 2 T and x 2 R . B The function is continuously di erentiable and satis es t > 0 for all t 2 T . Furthermore we assume C X 0 is indep endent of W and 2 E jX 0 j K 4 for some constant K > 0. 4 The prop erties A and C ensure the existence of a pathwise unique strong solu- tion of equation 1 . We use n h; to denote the total number of steps and write e X ; a; ;X 0 instead of e X . With kk we denote the L -norm of real-valued p p functions on T . Theorem 1. Assume that A { C hold for equation 1 . Then p 1=2 h b 6 k k lim n h; e X ;a;;X 0 = 1= 1 h!0 b for the Euler approximation with discretization 3 . The Euler approximation X with n constant step-size =1=n 4 k +1 k yields p 1=2 b lim n e X ;a;;X 0 = 1= 6 k k : n 2 n!1 3 Hence it is not ecient to discretize equidistantly: taking 3 instead of 4 reduces the error roughly by the factor k k =k k for the same numb er of steps. Even amuch 1 2 h b stronger optimality prop erty holds for the metho d X . This metho d is asymptotically optimal for all equations 1 among all metho ds that use values of W at a nite number of p oints. See Theorem 2 and Remark 2. 2=3 3 Remark 1. The term h in 3 only matters if h> . For small values of k k still a reasonably small step-size is needed to get a go o d approximation. The particular 2=3 choice h based on error estimates on intervals [ ; ]. k k +1 In the case of arbitrary scalar equations dX t = a t; X t dt + t; X t dW t ; t 2 T; 5 we prop ose the natural generalization of 3 , given by 2=3 h b = + min h ; h= ; X ; 6 k +1 k k k as an adaptive step-size control. Simulation studies indicate that the discretization 6 is still sup erior to an equidistant discretization. The asymptotic analysis of the step-size control 6 for equations 5 is an op en problem. 3 Lower Error Bounds We present lower b ounds that hold for every n-p oint metho d. For that purp ose we drop all restrictions on the available information ab out a and . We x a and , and we consider the corresp onding equation 1 . By an n-p oint metho d we mean an arbitrary metho d X , that is based on a realization of the initial value X 0 and on n n observations of a tra jectory of W . Such a metho d is de ned by measurable mappings k : R ! T k for k =1;:::;n and n+1 : R ! L T : n 2 The mapping determines the observation p oint in step k in terms of the previous k evaluations. A pathwise approximation is computed according to X = X 0 ;Y ;:::;Y ; n n 1 n 4 where Y = W X 0 , and 1 1 Y = W X 0 ;Y ;:::;Y k k 1 k 1 is the observation in step k 2. The quantity e n; a; ;X 0 = inf e X ;a;;X 0 n X n is the minimal error that can b e obtained by n-p oint metho ds for equation 1 . Theorem 2. Assume that A { C hold for equation 1 . Then p 1=2 lim n e n; a; ;X 0 = 1= 6 k k : 1 n!1 Remark 2. Due to Theorem 1 and 2 the Euler approximation with adaptive step-size control 3 is asymptotically optimal for every equation 1 . Apart from Clark and Cameron 1980 , there is a lack of pap ers dealing with lower b ounds. Clark and Cameron derive lower and upp er b ounds for n-p oint metho ds that are based on the equidistant discretization 4 . Theorem 2 provides for the rst time a lower b ound for arbitrary metho ds which use discrete observations of a Brownian path. References Clark, J. M. C. and Cameron, R. J. 1980 . The maximum rate of convergence of discrete approximations. In Stochastic Di erential Systems B. Grigelionis, ed. 162{171. Lecture Notes in Computer Science 25. Springer, Berlin. Hofmann, N., Muller-Gron bach, T. and Ritter, K. 1998 . Optimal Approximation of Sto chas- tic Di erential Equations by Adaptive Step-Size Control. submitted for publication in Math. Comp. Klo eden, P. and Platen, E. 1995 . Numerical Solution of Stochastic Di erential Equations. Springer, Berlin. Milstein, G. N. 1995 . Numerical Integration of Stochastic Di erential Equations. Kluwer, Dordrecht. Talay, D. 1995 . Simulation of sto chastic di erential systems. In Probabilistic Methods in Applied Physics P. Kr ee, W. Wedig, eds. 54{96. Lecture Notes in Physics 451, Springer, Berlin. 5 INFINITE DIVISIBILITY REVISITED ZBIGNIEW J. JUREK Abstract. For the class ID of all in nite divisible measures on a Banach space, weintro duce a family of convolution semi-groups that sum up to the class ID. Each semi-group in question is given as a class of distributions of some random integrals with resp ect to some Levy pro cesses. At the same time, it is de ned as a class of limit laws. The aim of this talk is to show usefulness of random integrals in clas- sical limit distributions theory. Our integrals will b e pathwise integrals with resp ect to some L evy pro cesses , i.e., Z Z def t=b f t dY t = f t Y t j 1 Y t df t ; t=a a;b] a;b] where f is a real-valued function with b ounded variation and Y is a L evy pro cess stationary and indep endent increments with cadlag paths. Moreover, Z Z def f t dY t ; a:s: f t dY t = lim b"1 a;b] a;1 By ID we denote the class of all in nitely divisible probability mea- sures. It coincides with laws of Y 1 , where Y is an arbitrary L evy pro cess . We will de ne a ltration of the ID into increasing convolu- tion semi-groups U ; 2 R Namely: DEF1. We say that 2U if there exists a sequence of indep endent L evy pro cesses t ;j =1; 2;::: ; such that j 1 L + + ::: + n = ; as n !1; 1 2 n n i.e., one evaluates the distributions of the arithmetic average at times n . Theorem 1 a For 2 < 1, Z 2 a symmetric 2U i = L tdY t ; 0;1 where is symmetric stable measure with exp onent and Y is a unique L evy pro cess such that E [jjY 1 jj ] < 1. b for 1 < < 0; 2 U i 2 holds with as strictly stable measure with exp onent . 1 2 ZBIGNIEW J. JUREK R R s c 2U i = L e dY s =L tdY ln t 0 0;1 0;1 where Y is a unique L evy pro cess with E log 1 + jjY 1 jj < 1. R d For > 0; 2U i = L tdY t 0;1 where Y is an arbitrary L evy pro cess . [To the L evy pro cesses in the ab ove random integrals we refer as the BDLP, i.e., background driving L evy pro cesses for the corresp onding distributions.] Remark. 1 If U has non-degenerate distribution then 2. 2 U consist of Gaussian distributions. 2 3 If then U U . 1 2 1 2 4 Each U is a closed convolution subsemigroup. 5 U = L, where L stands for L e vy class L distributions also called 0 self-decomp osable distributions . Corollary. A ltration of the class ID is: [ U = ID n n for any sequence " +1: [The bar means closure in weak top ology]. n Examples. 1 Comp ound Poisson distributions are in ID but they are not in class U = L of selfdecomp osable distributions. 0 2 Stable laws are in L. 3 Comp ound geometric laws waiting time for the rst success but not the momentofthe rst success are not in L. 4 Gaussian distributions, Student t-distribution, Fisher F -distribu- tions, generalized hyp erb olic distributions are in L; [3]. For r>0 let us de ne the shrinking op eration U as follows: U x =0, r r x otherwise. In case of x 2 R , if jjxjj r jjxjj note that x U x = c x is the censoring at the level r>0. Also, for r r x>0; U x is the pay-o function in some nancial mo dels, but here r we are interested in the following. DEF2. A measure is said to b e s-selfdecomp osable s here indicates s-op erations U , if is a limit of the following sequence r U X + U X + ::: + U X + x ; r 1 r 1 r n n n n n where X ;X ;::: are indep endent and the triangular array U X ; j = 1 2 r j n 1; 2;::: : n 1; is the uniformly in nitesimal. Theorem 2. is s-selfdecomp osable i 2U . 1 In the lecture we will discuss continuity of the random integral map- pings given via the integrals in Theorem 1. Remark. From random integrals easily follow characterization in terms of Fourier transformations. This allows to avoid the metho d INFINITE DIVISIBILITY REVISITED 3 of extreme p oints Cho quet's Theorem as used by D.Kendall, S. Jo- hansen or K. Urbanik. It seems that one might argue that random integral, as a metho d of describing limiting distributions, are more \natural or probabilistic" than the Fourier transform. Since the \random integral representations" as in Theorem 1 were successfully used for some other classes of limit laws, it led to the fol- lowing: Hyp othesis. Each class of limit class derivedfrom sequences of inde- pendent random variables is an image of some subset of ID via same random integral mapping. References [1] Z. J. Jurek, Random integral representations for classes of limit distributions similar to L evy class L , Probab. Theory Related Fields, 78 1988 , pp. 473{490. 0 [2] , Random integral representations for classes of limit distributions similar to Levy class L . III, in Probability in Banach spaces, 8 Brunswick, ME, 1991 , 0 Birkhauser Boston, Boston, MA, 1992, pp. 137{151. , Selfdecomposability: an exception or a rule?, Ann. Univ. Mariae Curie- [3] Sklo dowska Sect. A, 51 1997 , pp. 93{107. [4] Z. J. Jurek and J. D. Mason, Operator-limit distributions in probability the- ory, John Wiley & Sons Inc., New York, 1993. A Wiley-Interscience Publication. [5] Z. J. Jurek and B. M. Schreiber, Fourier transforms of measures from the classes U ; 2 < 1, J. Multivariate Anal., 41 1992 , pp. 194{211. Institute of Mathematics, University of Wroclaw, Pl. Grunwald zki 2/4, 50-384 Wroclaw, Poland E-mail address : [email protected] GRAPHICAL REPRESENTATION FOR INTERACTING PARTICLE SYSTEMS BO MARKUSSEN An Interacting Particle System is a Markov jump-pro cess on the S space X = W of con gurations on an in nite site-space S , where each particle takes a state from a phase-space W . The so-called graphical representation gives an explicit construction as a function of in nite many Poisson pro cesses. For this construction to be well-de ned we need to imp ose a condition on the interactions between the particles such that for every time-p oint T the site-space splits into nite regions which do not interact b efore time T . The pro of of the well-de nedness uses the Ito formula for counting pro cess integrals. Inst. for Matematiske Fag, K benhavns Universitet, Universitetsparken 5, 2100 K benhavn , Danmark E-mail address : [email protected] 1 p-variation and integration of sample functions of sto chastic pro cesses R. Norvaisa Institute of Mathematics and Informatics Akademijos 4, Vilnius 2600, Lithuania Let h b e a real{valued right{continuous function on a closed interval [a; b]. Consider the re nement{Riemann{Stieltjes integral equation Z y a F y = 1+ RRS F dh; a y b; 1 a for a function F . Here and b elow for a regulated function f on [a; b], we write b b b := f b ; x :=f x :=f x+ := lim f z if a x + z x + + a a f x := f x :=f x := lim f z if a z "x Let W = W [a; b] b e the set of all functions on [a; b] with b ounded p-variation. If h 2W p p p for some 0 p pro duct integral n y Y h h P y = P y := 1 + dh := lim [1+h x h x ]; i i 1 a " a i=1 where the limit exists under re nements of partitions = fx : i =0;:::;ng of [a; y ]. i Several asp ects leading to linear integral equation 1 were discussed in Section 7.4 of [2]. The inde nite pro duct integral is de ned with resp ect to any function h having b ounded p- variation for some 0 h may ask what integral equation satis es the inde nite pro duct integral P when h is not right{continuous? In this case wehave that + h h + P y = P y h y if a y h h P y = P y h y if a< y b h see Lemmas 5.1 and 5.2 in [1] . The di erentvalues of P on the right side maylookun- usual, so that ab ove relations should b e considered as an evolution on the extended time scale fa; a+;:::;y ;y;y+;:::;b ;bg. Because the RRS integral need not exist when jumps of an integrand and integrator app ear at the same p oint on the same side, wehave to replace it by more general integrals discussed in [2]. For example, consider the re nement{Young{Stieltjes integral equation Z y 0 F F y = 1+ RY S dh; 0 y 2; 2 0 1 2 with resp ect to function h de ned by 8 > 0 if 0 x< 1 < 1=2 if x =1 h x := 3 > : 1 if 1 Then 8 > 1 if 0 x<1 < h P x = 1+ 1=2 if x =1 > : 2+ 1=4 if 1 and Z y 1 if 0 x< 1 0 h h 1+ RY S P dh = 6= P y for y 1. 2 if 1 x 2 0 h Thus P do es not satisfy 2 . Let f and g b e regulated functions on [a; b]. De ne the Left Young integral, or the LY integral, by Z Z b b X a b + + LY gdf := RRS g df +[g f ] a + g f + a a a;b provided the RRS integral exists and the sum converges absolutely.Wesay that g is LY integrable with resp ect to f on [a; b]. The LY integral was de ned in [1, Relation 3.44 ] for Banach algebra valued functions to provide a suitable extension of Duhamel's formula for pro duct integrals. To compare the LY integral with the RY S integral we recall the CY integral de ned byL.C.Young [5] as follows: Z Z b b X a b + g f +[ g f ] b CY gdf := RRS g df +[g f ] a + + a a a;b provided the RRS integral exists and the sum converges absolutely. By Prop osition 3.17 of [1], or by Theorem 6.20 of [2], if the RY S integral exists then so do es the CY integral, and the two are equal. It is instructivetocheck that the inde nite pro duct integral with resp ect to the function h de ned by 3 satis es the Left Young integral equation Z Z y y X 0 y h h h + h + h 1+ LY P dh = 1+ RRS P dh +[P h] 0 + P h = P y + 0 0 0;y for 0 y 2. De ne the Right Young integral, or the RY integral, by Z Z b b X b a + g df g f +[g f ] b gdf := RRS RY + a a a;b provided the RRS integral exists and the sum converges unconditionally.Wesay that g is RY integrable with resp ect to f on [a; b]. Next follows from Theorem on Stieltjes integrability ofL.C.Young [5, p. 264]. 3 Theorem 1. Let g 2W [a; b] and f 2W [a; b] for some p; q > 0 such that 1=p +1=q > 1. p q b b Then the integrals LY s gdf and RY s gdf are de ned. a a The following statement is proved in [1, Theorems 5.21 and 5.22] for Banach algebra valued functions. Theorem 2. Let f 2W [a; b] with 0 p x f P x := 1 + df ; a x b; a a exists and is the unique solution in W [a; b], for any r p such that 1=p +1=r > 1, of the r LY integral equation Z x F x = 1+ LY Fdf; a x b: a Similarly, the inde nite product integral b f P y := 1 + df ; a y b; b y exists and is the unique solution in W [a; b], for any r p such that 1=p +1=r > 1, of the r RY integral equation Z b G y = 1+ RY Gdf; a y b: y A di erent pro of of this statement based on the chain rule formula Theorem 6 b elow , is due to Mikosch and Norvaisa [3]. In the same manner they also solved the non-homogeneous linear LY and RY integral equations. For illustration we formulate several prop erties of the LY and RY integrals. Their pro ofs are given in [4]. Theorem 3. Let g , f beregulated functions on [a; b], and let a c b.For A = LY or RY , b c b gdf both exist, and then gdf and A s gdf exists if and only if A s A s c a a Z Z Z b c b A gdf = A gdf + A gdf: a a c Next is the integration by parts formula for the LY and RY integrals. Theorem 4. Let g and f beregulated functions on [a; b]. If either of the two integrals b b LY s gdf and RY s fdg exists then both exist, and a a Z Z b b LY gdf + RY fdg = f b g b f a g a : a a De ne the inde nite LY and RY integrals by Z Z x b x := LY gdh and y := RY g dh; a y resp ectively for x; y 2 [a; b]. Next is the substitution rule for the LY and RY integrals. 4 1 1 Theorem 5. Let h 2W [a; b] and f; g 2W [a; b] for some p; q > 0 with p + q > 1. Then g p q and fg are LY integrable with respect to h, f is LY integrable with respect to the inde nite LY integral , and Z Z b b LY fd = LY f g dh : a a Similarly, g and fg are RY integrable with respect to h, f is RY integrable with respect to the inde nite RY integral , and Z Z b b RY fd= RY f g dh : a a It^o's formula gives a sto chastic integral representation of a comp osition of a smo oth function and a Brownian motion. We prove the Left Young integral representation for a comp osition f of a smo oth function and a function f having b ounded p-variation with p 2. Recall that almost every sample function of a Brownian motion has b ounded p-variation for each p>2. Besides its applications to sample functions of sto chastic pro cesses the chain rule formula extends Theorem 1 of L. C. Young to the case when 1=p +1=q = 1. Notice that this extension concerns the existence of integrals with integrands of a sp ecial form. Let d b e a p ositiveinteger and let b e a real-valued function on a d-dimensional cub e d d [s; t] := [s; t] [s; t]. We write 2H [s; t] if satis es the condition: 1 is 1;0 @ d 0 di erentiable on [s; t] with continuous partial derivatives u := u for l =1;:::;d.For l @u l d 0 d each 2H [s; t] , let K := 2 max supf u : u 2 [s; t] g. Also, for 2 0; 1], we 1;0 0 1ld l d write 2H [s; t] if, in addition to condition 1 , satis es the condition: 2 there is a 1; nite constant K such that the inequality d X 0 0 v j K ju v j u max j k k l l 1ld k =1 d holds for all u = u ;:::;u ;v = v ;:::;v 2 [s; t] . 1 d 1 d The following statement provides the LY integral representation of a comp osition under the b oundedness of p-variation condition. d Theorem 6. For 2 [0; 1], let f = f ;:::;f : [a; b] 7! s; t bea vector function with 1 d d coordinate functions f 2W [a; b] for l =1;:::;d, let 2H [s; t] and let h bea l 1; 1+ regulated function on [a; b]. Then the equality Z Z d b b X 0 LY hd f = LY h f df l l a a l=1 d d X X X X 0 + 0 + f f ]; f f ]+ h[ f + h [ f l l l l l=1 l=1 [a;b a;b] holds meaning that al l d +1 integrals exist provided any d integrals exist, and the two sums converge unconditional ly. We plan to discuss applications of these results to sample functions of sto chastic pro cesses. 5 References [1] Dudley, R. M. and Norvaisa, R. 1998 . Pro duct integrals, Young integrals and p-variation. Lect. Notes in Math., Springer. to app ear [2] Dudley, R. M. and Norvaisa, R. 1998 . An intro duction to p-variation and Young integrals. Lecture Notes, 1, MaPhySto, University of Aarhus, Denmark. [3] Mikosch, T. and Norvaisa, R. 1998 . Sto chastic integral equations without probability. Bernoul li. to app ear [4] Norvaisa, R. 1998 . p-variation and integration of sample functions of sto chastic pro cesses. Preprint. [5] Young, L. C. 1936 . An inequality of the Holder typ e, connected with Stieltjes integration. Acta Math. Sweden , 67, 251-282. INDEPENDENCE OF MULTIPLE STOCHASTIC INTEGRALS JAN ROSINSKI The problem of indep endence of multiple sto chastic integrals with resp ect to in nitely divisible random measures is investigated. It is remarked that multiple Ito-Wiener integrals are indep endent if and only if their squares are uncorrelated. This criterion breaks down for multiple Poisson integrals. The necessary and sucient condition for the indep endence of mul- tiple integrals of arbitrary not necessary equal orders with resp ect to a symmetric stable random measure is established. To this aim the pro duct formula for multiple sto chastic integrals is studied through se- ries representations. The knowledge of the tail asymptotic b ehavior for multiple stable integrals and their pro ducts plays the crucial role in our metho d, its role is similar to Ito's isometry for Ito-Wiener integrals. University of Tennessee, Department of Mathematics, Knoxville, TN 37996 USA E-mail address : [email protected] This is based on jointwork with Gennady Samoro dnitsky. 1 FORWARD AND BACKWARD CALCULUS WITH RESPECT TO FINITE QUADRATIC VARIATION PROCESSES by Francesco RUSSO and Pierre VALLOIS 1 Sto chastic integration via regularization This pro ject which is carried on since 1991 in several pap ers, see [RV1; RV2; RV3; RV4; RV5; RV6; ER; ERV], but also in [W1;W2;W3] and [RVW]. It covers the topic of the following two talks in Aarhus. F. Russo: Calculus with resp ect to a nite quadratic variation pro cess. P. Vallois: Sto chastic calculus related to general Gaussian pro cesses and normal martingales. The aim of the pro ject was to develop a sto chastic calculus which has essen- tially four features. 1. It b elongs to the context of \pathwise" sto chastic calculus. To this extent, one of the basic reference is the article of H. Follmer [F] , continued for instance in [Be] but also by Dudley and Norvaisa, see contributions in this volume. Follmer's metho d is based on integrator discretization, ours works out regularization to ols. 2. One asp ect of our approach is the particular simplicity. In fact, many rules are directly derived using rst year calculus arguments and nite Taylor expansions, uniform continuity and so on. 3. Our approach establishes a bridge b etween anticipating and non-anticipating integrals, see for instance [NP], [N]. Concerning anticipating calculus, there are two main techniques at least as far as the integrator is a Brownian motion: Skoroho d integration which is based on functional 1 analysis, see for instanse [NP], [N], and enlargement of ltration see e.g. [J] to ols which are based on classical sto chastic calculus metho d. Our integrals help to relate those two concepts. 4. Our theory should allowtogobeyond semimartingales. Our aim was to understand what could b e done when integrators are just Gaussian pro- cesses, convolution of martingales, as fractional Brownian motions and Dirichlet typ e pro cesses sum a lo cal martingale and a zero-quadratic variation pro cesses . All the pro cesses will b e supp osed to be continuous for simplicity . Exten- sions to the jump case are done in [ERV]. Let X , Y be continuous pro cesses. We set t t0 t t0 Z Z t t X X s+" s ds Y Yd X = ucp lim s + "!0 " 0 0 Z Z t t X X s s " _0 + Yd X = ucp lim Y ds s + "!0 " 0 0 Z Z t t X X s+" s " _0 o ds Yd X = ucp lim Y s + "!0 2" 0 0 Z t 1 [X; Y ] = ucp lim X X Y Y ds t s+" s s+" s + "!0 " 0 if previous quantities do exist; ucp denotes the uniform convergence in prob- abilityon each compact. R R R t t t + o Yd X resp. Yd X; Yd X is called the forward resp. backward, 0 0 0 symmetric integral of Y with resp ect to X provided that those integrals do exist. [X; Y ] is called the covariation of X and Y . [X; X ] is also called quadratic variation of X and X . If [X; X ] exists X will b e said to b e a nite quadratic variation pro cess. In this case, it is an increasing pro cesses, b eing a limit of increasing pro cesses; therefore it is a classical integrator in the sense of measure theory of IR +. 1 m Let X ;:::;X be some pro cesses. 2 1 m De nition fX ;:::;X g are said to have all their mutual brackets if i j [X ;X ] exists 8 1 i; j m. In this case : i j i j i i i j j j [X + X ;X + X ]=[X ;X ]+2[X ;X ]+[X ;X ] and i j [X ;X ] are lo cally with b ounded variation: Prop erties We rst state some elementary prop erties which directly follow from the de nition and ordinary calculus. Z Z t t + Y d X . Y d X 1. [X; Y ] = t 0 0 Z Z t t + Z Y d X + Y d X t o 0 0 Y d X = 2. 2 0 ^ 3. For T > 0we set X = X . Then t T t Z Z T t + ^ ^ Y d X Y d X = T t 0 1 4. Let f; g 2 C IR , fX; Y g having all their mutual brackets. Then ff X ;g Y g have the same prop erty and Z t 0 0 [f X ;g Y ] = f X g Y d[X; Y ] : t s s s 0 2 5. It^o formula. Let f 2 C and X a nite quadratic variation pro cess. Then : Z Z t t 1 0 00 f X = f X + f X d X f X d[X; X ] t 0 s s 2 0 0 Z t 00 0 f X d[X; X ] = [f X ;X] : t 0 3 6. If [X; X ] exists, [Y; Y ] = 0 then fX; Y g have all their mutual brackets and [X; Y ]=0. 7. Integration by parts. Z Z t t + X Y = X Y + Xd Y + Yd X t t 0 0 0 0 Z Z t t = X Y + Xd Y + Yd X +[X; Y ] 0 0 t 0 0 2 Examples Even if the theory develop ed until now is very simple, it would be empty, if we cannot provide a rich enough class of examples. 1. If X has b ounded variation Z Z Z t t t + YdX Yd X = Yd X = 0 0 0 The fact that the rst and second memb ers equal the third is a conse- quence of the de nition of forward and backward integrals, of Fubini theorem and of the dominated convergence theorem. 2. Let X , Y two semimartingales with resp ect to some usual ltration i F ;H some adapted pro cesses, i = 1; 2. Then [X; Y ] is the usual t 1 2 bracket of M and M , those pro cesses b eing the lo cal martingale part Z t 1 of X and Y . Then H d X coincides with the classical It^o integral 0 and Z Z Z t 1 1 2 2 1 2 1 2 dM ; H dM H = H d[M ;M ]: H 0 0 0 t We remark that that prop erty will play in the sequel the role of a de nition of a "go o d" nite quadratic variation pro cess. 3. When X is a Brownian motion B , under suitable assumptions on a R pro cess H , it is p ossible to relate the Skoroho d integral H@B with 0 4 R 2 Hd B . If H 2D L [0;T] , the forward and backward integrals 2;1 0 D b eing Wiener-Sob olev spaces of Malliavin-Watanab e typ e, 2;1 2 2 2 DH = D H : r;s 2 [0;T] 2 L [0;T] : r s then Z Z t t Hd B = HB + Tr DH t 0 0 where Z Z t r 1 dr ds D H Tr DH t = lim r s "!0 " 0 r " _0 Z Z Z t 1 2 1 2 H B ; H B = H H ds: s s 0 0 0 t For this see [RV1, SU, Z]. 4. Since the forward integral do es not see any ltration, it also coincides with any enlargement of ltration integral. Previous p oint relates pre- cisely the integral constructed by enlargement of ltrations and Skoro- ho d integral. 5. Dirichlet-Fukushima pro cess. A Dirichlet pro cess in the pathwise sense is a pro cess X of the form X = M + A where M is a F - lo cal martingale, and A is a zero t quadratic variation F -adapted pro cess. This concept has b een in- t spired by [Fu] who considered functions of a good stationary pro cess coming out from a Dirichlet form for which he could obtain the ab ove decomp osition. We observe that [X; X ]=[M; M ] 1 ~ ~ If f 2 C , Y = f X is again a Dirichlet pro cess with Y = M + A R t 0 ~ ~ ~ and M = f X + f X dM where A = f X M . Using t 0 t t t 0 the bilinearity of the covariation, prop erty d stated in section 1 ~ ~ and example 2 we easily obtain [A; A] 0. 6. Extended Lyons-Zheng pro cesses This example is treated in details in [RVW]. For simplicitywe consider here only pro cesses indexed by [0; 1]. 5 ^ For X = X t , t 2 [0; 1] ; we set X t =X 1 t : Let F = F , H = H be two ltrations. t t t2[0;1] t2[0;1] ^ ^ If Y t Y 0 is F -adapted and Y t Y 0 is H -adapted then we say that Y is weakly F ; H -adapted De nition A continuous weakly F ; H -adapted pro cess X is called a F ; H - LZ i i pro cess if there are M = M t , t 2 [0; 1] , i = 1; 2, V = V t , t 2 [0; 1] , such that 1 1 2 M + M +V X = 2 and the following conditions are satis ed: 1 a M is a lo cal F -martingale with 1 M 0 = 0: 2 ^ b M is a lo cal H -martingale with 2 M 1 = 0: c V is a b ounded variation pro cess. 1 2 d M M is a zero quadratic variation pro cess. Remark 2.1 Let X be an F ; H -LZ pro cess. 1 1 1 2 2 1 [X; X ]= [M ;M ]+[M ;M ] : 2 2 A time reversible semimartingale is a LZ pro cess with resp ect to the natural ltrations. 3 If Y is a F ; H -weakly adapted pro cess then we de ne the LZ- symmetric integral by Z Z Z Z t 1 t t 1 1 1 2 ^ ^ Y dX = Y d M Y d M + Y dV : 2 2 0 1 t 0 0 1 2 If [Y; M M ]=0 then Z Z t t Y dX = Yd X 0 0 6 7. Delayed pro cesses Let > 0; S a F -semimartingale and X a F -adapted t t t t _0 pro cess. Then Z Z Z t t t + [X; S ] 0, Xd S = Xd S = XdS 0 0 0 8. Convolution of martingales. This example has b een explicitely discussed in [ER]. Let M b e a martin- Z t gale and g b e a continuous real function such that X = g t s dM : t s 0 If [g u ;g] exists, for any u 0, then [X; X ] exists and can be explicitely calculated. 9. Substitution formulae. d d Let X t; x ;t 0;x 2 IR , Y t; x , t 0, x 2 IR semimartin- d gales dep ending on a parameter x, H t; x ;t 0;x 2 IR predictable pro cesses dep ending on x . We make assumptions of Garsia-Rudemich-Rumsey typ e. Then Z Z t t H s; Z d X s; Z = H s; x dX s; x x=Z 0 0 [X ;Z ;Y ;Z ] = [X ;x ;Y ;x x=Z Those formulas are useful for proving existence results for SDE's driven by semimartingales with anticipating initial conditions. 10. Gaussian case. Let X be a Gaussian pro cess such that t m t =E X ; t K s; t =Cov X ;X = E X X m s m t s t s t Provided [m; m] = 0, it is enough to supp ose that X is a mean-zero pro cess. 7 Prop osition 2.2 We supp ose : 1 K 2 C + where + = f s; t : 0 s tg We denote D K s; s+ the restriction of D K on the diagonal. 2 2 Then [Y; Y ] exists, it is deterministic and it equals t Z t a t =K t; t K 0; 0 2 ds D K s; s+ : 2 0 Particular cases: - X = B : Brownian motion. K s; t = s ^ t Z t ds D K s; s+ = 0 a t =t: 2 0 -Fractional Brownian motion. 1 K s; t =K s; t = jsj + jtj js tj 2 a t =0 1 < <2: In such a case it is p ossible to show that the variation of such a pro cess equals t. We recall that a sto chastic integral and calculus for fractional Brownian motion has b een for instance develop ed by [DU] and [Z1;Z2]. Concerning forward integration, wehave the following. Supp osing that 1 K is of class C on , we set + Z Z 2 2 A f; f t = 2E [ 1 f X f X f u; v X + u; v X 1 u v 1 2 fuv tg u v + u; v X X + u; v gdudv , 1;2 u v ; ; and b eing four functions de ned explicitely through 1 2 1;2 K . Z t @K @K 2 A f; f t = E [ f X ds. s; s s; s 2 s @u @v O 8 A = A + A . 1 2 Theorem 2.3 a A f; f t 0 8 t 0, for every continuous and b ounded real function f . Z t X s + " X s b Supp ose that A f; f < 1 then f X ds, s " 0 Z t 2 converges, as " ! 0 ,inL to f X d X . + s s O c Moreover Z t 2 E f X d X = A f; f t s s 0 Z t f X d[X; X ] s = A f; f t E s 2 0 Z Z @K t t s; s r @u ds: E f X d X = E [f X X ] s s s s K s; s 0 0 11. Normal martingales as integrators The basis for developing forward integration with resp ect to a normal martingale has b een develop ed in [RV5]. Supp ose M is a normal martingale of it is lo cally square integrable martingale and hM; M i t ;t 8 t 0, hM; M i b eing the dual pro jection of the classical bracket [M; M ] ref. [DMM] . Let X ; t 2 [0; 1] b e a square integrable pro cess, which means t Z 1 2 2:1 E X ds < 1: s 0 We supp ose X 2:2 X t = I f t; ; 8 t 2 [0; 1] n n n0 where I f t; is the n multiple iterated sto chastic integral of f t; n n n with resp ect to M . f t; is a square integrale and symmetric function n n . de ned on IR + 9 If M has the chaos decomp osition prop erty, every square integrable random variable has such a decomp osition. In particular 2.1 implies that 2.2 holds. We set X the Skoroho d integral of X with resp ect to M : X ~ X = I f ; n+1 n n0 2 ~ provided the series converges in L , f denoting the symmetrization of n f considered as a function of n +1 variables. n Z t In the Brownian case, Xd M is the sum of X plus a trace term. 0 The pro of is based on the following identity: Z 1 g 2:3 I f I g =I f g +nI f t; g t dt : n 1 n+1 n 1 0 This prop erty can b e generalized to M , namely Z 1 1 g I f I g = I f g +n I f t g t d [M; M ] t n 1 n+1 n 1 0 1 where I f t ; t 0 is the unique element verifying n 1 Z Z 1 1 1 2 1 E I f t d[M; M ] t = I f t dt n 1 n 1 0 0 2 = n! kf k : n 2 L IR + Then Z 1 X dM = X + Tr X 2:4 0 Z 1 X 1 ~ Tr X = d[M; M ] s n I f s ;s; n n 1 0 n Z s 1 1 1 ~ I f t; s dt: I f s ;s; = lim n n n 1 n 1 "!0 " s " 2 if the two series converges in L . 10 3 Generalized It^o pro cesses. For a one-dimensional pro cess X , the existence of the quadratic variation is equivalent to the validity of It^o formula in the sens of prop erty 4, section 1. This is illustrated by the following lemma. Lemma 3.1 Let X be a pro cess. X is a nite quadratic variation pro cess if and only if Z 1 3:1 g X d X exists 8 g 2 C IR : 0 Pro of 2 0 Let G 2 C such that G = g . Using It^o formula, Z Z t t 1 0 g X d X = G X G X g X d[X; X ]: t 0 2 0 0 Conversely if 3.1 holds, Z Z t t + [g X ;X] = g X d X g X d X exists: t 0 0 Taking g x = x, X is proven to be a nite quadratic variation pro cess. 2 This result motivates the n-dimensional case. 1 n 1 n Let fX ;:::;X g having all their mutual brackets. X ;:::;X is called generalized It^o pro cess if Z i 1 n 3:2 g X d X exists; 8 g 2 C IR : 0 This means g X 2 I where I is the class of pro cesses Z such that X X t Z i 3:3 Zd X exists; 1 i n: 0 11 1 n 2 Prop osition 3.2 Let X = X ;:::;X an It^o pro cess, ' 2 C , Y = 1 n ' X . Let Z another pro cess. If Zg X 2 I ; 8 g 2 C IR , then Z 2 I t X Y and 3:4 Z Z Z n n t t t X X 1 2 i j i @ Z ' X d[X ;X ] @ ' X d X Zd Y = Z s s s i s s ij s 2 0 0 0 i;j =1 i=1 Remark 3.3 1 i If Z = f X , f 2 C then 3.4 can b e applied. ii 3.4 formally can be expressed as : n n X X 1 i 2 i j 3:5 d Y = @ ' X d X @ ' X d[X ;X ] s i s s s s ij 2 i=1 i;j =1 1 n De nition A pro cess X = X ;:::;X will b e called vector It^o pro cess if Z Z Z t i j i j f X d X ; g X d X = f X g X d[X ;X ] : s s s 0 0 0 t Remark 3.4 If n =1,any nite quadratic variation pro cess is a vector It^o pro cess. 4 Sto chastic di erential equations Let a nite quadratic variation pro cess. We are interested by t t0 4:1 d X = X d ; X = t t t 0 2 2 C , any random variable. b 12 Remark 4.1 If is a semimartingale even for the case Lipschitz there t is a solution X of t t0 Z t 4:2 X = + X d : t s s 0 Pro of We apply substitution metho d, see section 2. 2 However a serious uniqueness result was missing. The problem is probably not well-p osed in the to o large class of all pro cesses for which the equation makes sense. So we have the following alternative. a either we restrict the class of pro cesses X , b or we mo dify a little bit the sens of solution; of course it will b e necessary to include the classical cases. We have chosen this second solution. De nition A pro cess X is solution of 4.1 if t a X; isavector It^o pro cess. b For every Z = ' X ; , ' regular, t t t Z Z t t Z d X = Z X d s s s s s 0 0 Remark 4.2 If is a semimartingale, then the solution which is obtained t by substitution ful lls a and b . 2 Prop osition 4.3 Let be a nite quadratic variation pro cess, 2 C . t b There is a unique solution X to problem 4.1 . Pro of We give an idea of the uniqueness pro of, since the existence involves very similar arguments. 13 We apply here the metho d develop ed for classical one-dimensional di usions by [Do] and [Su]. We consider the deterministic ow F : IR IR ! IR + solution of 9 8 @F = < r;x = F r;x : @r ; : F 0;x = x: 2 2 F is C and there is H : IR IR ! IR of class C such that + 4:3 F r;H r;x x; H r;F r;x x: We set Y = H ;X : t t t By It^o formula we have: Z Z t t @H @H 4:4 Y = Y + ;X d + ;X d X + BV t 0 s s s s s s @r @x 0 0 where BV is a b onded variation pro cess. Through 4.3 , we have @H @H r;x = x r;x @r @x ; X isavector It^o pro cess. Using Prop osition 3.2 the two sto chastic inte- grals app earing in 4.4 cancel. Then we solve pathwise 4.4 . Extensions are p ossible in the following cases: d X = X d + t; X d S t t t t t X = 0 when a S semimartingale, non-anticipating and is Lipschitz. t b S having b ounded variation, any random variable. t 14 5 Extended It^o formulae A particular class of generalized vector It^o pro cesses is constituted by pro- cesses X = X such that t t0 Z 5:1 Z g = g X d X 0 exist for any g 2 C IR : Prop osition 5.1 Z t 0 0 G X d X [G X ;X] 5:1 G X = G X t t 0 0 1 for any G 2 C IR : 2 Pro of 5.1 holds for G 2 C . Then Z Z t t X X X X s s " _0 s+" s ds; g ! g X ds 5:2 g ! g X s s " " 0 0 are continuous from C IR to the space C IR of continuous pro cesses equipp ed with the ucp top ology. The conclusion follows easily by Banach - Steinaus theorem and 5.1 . Example: X: time reversible semimartingale. Z Z T t + ^ ^ g X dX: g X d X = T t 0 0 We remark that [G X ;X] do es not need to b e of b ounded variation. o 0 If X is a Brownian motion B and f 2 C IR . Then [G X ;X] is of b ounded variation if and only if G X is a semimartingale. This is only p ossible if 0 G is di erence of convex functions. Therefore fG X ;Xg have not al l the mutual brackets. Prop osition 5.2 The following prop erties are equivalent. 1. [Z g ;Z g ] exists for any g 2 C IR : 15 2. [[g X ;X]; [g X ;X]] = 0; g 2 C IR : + + 3. [Z g ;Z g ] exists for any g 2 C IR : Z t 2 4. [Z g ;Z g ] = g X d[X; X ] . s s 0 Z t 2 + + 5. [Z g ;Z g ] = g X d[X; X ] . s s 0 Remark 5.3 Results in this direction have b een obtained by [RV4], for the case of time-reversible semimartingales; [BY] obtained a similar result for semimartingales where the bracket term is expressed by a generalized integral involving lo cal time. [FPS] for the case of Brownian motion, [BJ] for the case of elliptic di usions and [MN] for non-degenerate martingales, 1;2 . explore the case of f b eing in W loc Remark 5.4 If X = M + A, M b eing a time reversible semimartingale, then [g X ;X]; g 2 C IR exists and it has zero quadratic variation. Then Z g X d A 0 exists and it has zero bracket. Prop osition 5.5 Generalized It^o-Tanaka formula Let X = X be a nite quadratic variation pro cess. t t0 Z g X d X [g X ,X] exists for any increasing continuous function g if and only if 0 exists for any increasing continuous function g . Then Z t 1 0 0 [G X ;X] G X d X + G X =G X + t s s t 0 2 0 1 for any G 2 C IR : 16 Remark 5.6 If in section 4 has previous prop erties, it is p ossible to t relax the assumption under which the equation d X = X d t t t has a solution. REFERENCES. [BJ] Bardina,X., Jolis, M. An extension of It^o formula for elliptic di usion , 83-109 pro cesses. Sto chastic Pro cesses and their Applications. 69 1997 . [Be] Bertoin, J., Sur une int egrale p our les pro cessus a -variation b orn ee. Ann. Probab. 17 , n 4, 1521-1535 1989 . [BY] Bouleau, N., Yor, M., Sur la variation quadratique des temps lo caux de 1981 . certaines semimartingales. C.R. Acad. Sci. Paris, S erie I 292 [CJPS] E. C inlar, J. Jaco d, P. Protter, M.J. Sharp e, Semimartingales and Markov pro cesses. Z. Wahrscheinlicht, Verw. Geb., 54, 161-219 1980 . [DU] Decreusefonds, L., Ustunel, A.S., Sto chastic analysis of the fractional Brownian motion. To app ear : Journal of Potential Analysis : 1998. [DMM] Dellacherie, C., Meyer P. 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Sto chastics 33 Francesco RUSSO, Universit e Paris 13, Institut Galil ee, Math ematiques, Avenue Jean-Baptiste Cl ement, F - 93430 Vil letaneuse PierreVALLOIS, Universit e Henri Poincar e Nancy 1, D epartement de Math ematiques, Institut Elie Cartan, BP 239, F - 54506 Vanduvre-l es-Nancy Cedex 20 TOLERANCE TO ARBITRAGE AND THE STOP-LOSS START-GAIN STRATEGY DONNA M. SALOPEK We contruct an arbitrage opp ortunity in a frictionless sto ck market when price pro cesses have continuous sample paths of b ounded p-variation with p 2 [1; 2 . In addition, we review when the stop-loss start gain strategy is self nancing for the various price pro cesses. Donna M. Salopek, Dept. of Mathematics, P.O.Box 800, University of Groningen, NL-9700 AvGroningen, The Netherlands E-mail address : [email protected] 1 SOME MAXIMAL INEQUALITIES FOR FRACTIONAL BROWNIAN MOTIONS ESKO VALKEILA Abstract. We give an analogue of the Burkholder-Davis-Gundy inequalities for fractional Brownian motions. The pro of is based on kernel transformations, which transform a frac- tional Brownian motion to a Gaussian martingale and back. This allows us to use the martingale version of the Burkholder-Davis-Gundy inequality for the pro of. 1. Introduction 1.1. Fractional Brownian motions. We work in probability space ;F;P . A real H valued pro cess Z = Z is a fractional Brownian motion with Hurst index H , if it is a continuous Gaussian pro cess with stationary increments and has the following prop erties: i: Z =0. 0 ii: EZ =0 for all t 0. t 1 2H 2H 2H iii: EZ Z = t + s js tj for all s; t 0. t s 2 1 . The standard Brownian motion is a fractional Brownian motion with Hurst index H = 2 1 The pro cess Z is self-similar and ergo dic. It has p-variation index p = . H 1.2. A maximal inequality. For any pro cess X denote by X the supremum pro cess: X = sup jX j. s st t d H From the self-similarity it follows for the supremum pro cess Z that Z = a Z . Hence for at t any p>0 we have then the following result using self-similarity: Theorem 1.1. Let T > 0 be a constant and Z a fractional Brownian motion with Hurst index H . Then p pH 1.1 E Z = K p; H T ; T p where K p; H = E Z . 1 The value of the constant K p; H is not at our disp osal. 1991 Mathematics Subject Classi cation. 60H05, 60G15, 60J65. Key words and phrases. fractional Brownian motions, maximal inequalities. This is based on jointwork with A. Novikov. Supp orted by the research grant committee of the University of Helsinki. 1 2 ESKO VALKEILA 1.3. Burkholder-Davis-Gundy inequality for continuous martingales. Let N be a continuous martingale. Then the Burkholder-Davis-Gundy inequalities are: Prop osition 1.1. For any p> 0 and stopping time there exists constants c ;C > 0 such p p that p p=2 p=2 1.2 c E hN; N i E N C E hN; N i : p p For the pro of we refer to [RY, Theorem IV.4.1]. For the sp ecial case of standard Brownian motion W we have p p p 2 2 c E 1.3 E [ W : ] C E p p 2. Transformations 2.1. The Molchan martingale M . We give some recently obtained integral representa- tion b etween a fractional Brownian motion Z and the Molchan martingale M . T The integrals b elow can be de ned by integration by parts, where the singularities of the kernels do not cause problems, due to the Hölder continuity of the fractional Brownian motion Z see [NVV, Lemma 2.1] . : : c 1 Put m t; s = s t s for s 2 0;t and m t; s = 0 for s>t, where = H , C 2 s H : C = 1 1 H B H ; 2 2H 2 2 and 1 : c = ; 1 3 B H + ; H 2 2 where the b eta co e cient B ; for ; > 0 is de ned by : B ; = : + Prop osition 2.1. For H 2 0; 1 de ne M by Z t 2.1 M = m t; s dZ : t s 0 2 M C Z 2 2H = F and variance hM i = Then M is a Gaussian martingale, with F t . t 2 4H 2 2H For the pro of we refer to [NVV, Prop osition 2.1]. SOME MAXIMAL INEQUALITIES FOR FRACTIONAL BROWNIAN MOTIONS 3 2.2. From Molchan martingale to fractional Brownian motion. For later use denote by c the constant 2 s 2 C : c = : 2 2 4H 2 2H R R 1 1 : t t H H 2 2 s dZ . Then Z = dY . Moreover, by de nition s = Put Y s s s t 0 0 Z t 1 c H 2 M = dY ; t s t s C 0 and by [NVV , Theorem 3.2] Z t 1 H 2 Y =2H t s 2.2 dM : t s 0 3. Burkholder-Davis-Gundy inequality for fractional Brownian motions 3.1. The result. The following generalizes 1.3 for fractional Brownian motions. Theorem 3.1. Let be a stopping time. Then for any p > 0 and H 2 [1=2; 1 we have that p pH pH 3.1 c p; H E E Z C p; H E ; and for any p> 0 and H 2 0; 1=2 we have that p pH 3.2 c p; H E E Z ; where the constants c p; H ;C p; H > 0 depend only from parameters p; H . 1 , upp er b ound. We give the pro of of the right-hand side inequality in 3.2. Case H > 2 3.1 . R 1 t H 2 We have that Z = s dY . Use integration by parts to get the upp er estimate for Z : t s 0 Z 2t Y : t t For the pro cess Y use the representation 2.2 to get the estimate 3.3 : 4Ht M Y t t From these two upp er b ounds we derive the following upp er b ound 2 3.4 Z 8HM t : t t Note that 3.4 is valid for any stopping time : 2 Z 8HM : Hence for any p>0 we have that p p p 2 p 3.5 E Z 8H E M : H H H = > 1, and r = we get With Hölder's inequality, 3.5 with q = 2 2H 1 1 H 1 1 pr p 2 p 2 q p q r E M 3.6 : E M E 4 ESKO VALKEILA Apply now 1.2 to obtain 1 2 pr pr p p pH 2 E M c C E 3.7 = c C E : p p 2 2 To nish, note that the right hand side inequality in 3.1 for H > 1=2 follows from 3.6 , and : p p 3.7 with constant C = C p; H = 8H c C and C is the constant in the Burkholder- p p 2 Davis-Gundy inequality. 1 is similar to the pro of 3.3. Remarks. The pro of for the lower b ound in the case of H < 2 1 in subsection 3.2. The pro of of the lower b ound in the case of H > is longer and we refer 2 to [NV ] for the pro of. 1 We do not have an upp er b ound for the case H < . 2 References [NVV] Norros, I., Valkeila, E. and Virtamo, J. 1998 . An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli, to app ear. [NV] Novikov, A. and Valkeila, E. 1998 . On some maximal inequalities for fractional Brownian motions. Statistics & Probability Letters, to app ear. [RY] Revuz, D. and Yor, M. 1991 . Continuous Martingales and Brownian Motion. Springer, Berlin. Department of Mathematics, P.O. Box 4, FIN-00014 University of Helsinki, FINLAND E-mail address : [email protected] STOCHASTIC DIFFERENCE EQUATIONS, DISCRETE FOKKER-PLANCK EQUATION AND FINITE PATH-INTEGRALS IMME VAN DEN BERG We consider an appropriate discretisation of the Fokker-Planck equa- tion. Then the Feynman-Kac solution takes the form of a nite path- integral, along the paths of an asso ciated sto chastic di erence equation. In some sp ecial cases, such as the Black-Scholes equation, the path- integral b ecomes a simple or double Riemann-sum. The transition to continuity is made using a higher-order De Moivre-Laplace theorem, and the nonstandard notion of shadow. Rijksuniversiteit Groningen, P.O.Box 800, 9700 AVGroningen, The Netherlands E-mail address : [email protected] 1