QUASl-«MARTINGALES AND STOCHASTIC INTEGRALS flux-E: for flu Demo of Ph. D. MICRIGAN STATE UNIVERSITY Eanafid L. Fisk €963 THESE; This is to certify that the thesis entitled QUASI-HARTINGALES AND STOCHASTIC INTEGRALS presented by Donald L. Fisk has been accepted towards fulfillment of the requirements for Ph 0 D 0 degree in Phi ' OSOPhY fiw fins/A; ' Major professor Date AQQUSQL l963 0-169 LIBRARY Michigan State University ' MSU LIBRARIES “ V RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ABSTRACT QUASI-HARTINGALES AND STOCHASTIC INTEGRALS by Donald L. Fisk A quasi-martingale is defined to be a stochastic process, with time parameter the closed unit interval, which is decomposable into the sum of a martingale process and a process having almost every sample function of bounded variation. The first problem considered is that of obtaining necessary and sufficient conditions for a process to be a quasi-martingale. The main result is given in section 3 of Chapter II. Necessary and sufficient conditions are given for a process with almost every sam- ple function continuous to be a quasi-martingale with certain Speci- fied properties. The second problem considered is that of defining a stochastic integral with respect to a quasi-martingale process. The integral is defined as the probability limit of Riemann-Stieltjes type sums. Sufficient conditions for the existence of the integral are obtained in section 2, Chapter III! Section 3, Chapter III deals with the properties of the integral. Parallels are drawn here between the ordinary Riemann-Stieltjes integral and the stochastic integral. Particular emphasis is placed on transform properties of the integral. Donald L. Fisk The dominating technique Chapter II and Chapter III, is the use of random stapping times defined in terms of the process or processes under consideration. The most significant use of stopping times is in obtaining sequences of processes so that each process in the sequence has a certain specified property (for example each process in the sequence may be uniformly bounded or each process .in the sequence may be almost surely sample equi-continuous) and the sequence of processes converge to the original process in the sense that there is eventual equality of almost every sample function- QUASI-NARTINGALES AND STOCHASTIC INTEGRALS By Donald L. Fisk A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics l963 i o .1," w ACKNOWLEDGMENTS The people to whom one is indebted in acquiring an education are indeed numerous. It is hoped these people will accept my sincere thanks. The time and effort expended by the writer's major professor, Dr. Herman Rubin, in developing the thesis can- not be overemphasized and are greatly appreciated. Special thanks are also due my parents, without whose assistance this would not have been possible, and to my wife and children who have had to sacrifice much. I would also like to thank the United States Depart- ment of Health, Education and welfare for their financial assistance in the form of a National Defense Education Act Fellowship, and also the Department of Statistics, Michigan State University, and the Office of Naval Research for their financial support during the writing of this thesis. ii TABLE OF CONTENTS Chapter I - Preliminary Discussion Chapter II Section 1: Random Stapping Section 2: Decomposition Theorem Section 3: Rain Decomposition Theorem Section A: Particular Results ,Chapter III - Stochastic Integrals Section I: General Discussion Section 2: Definition of the Integral Section 3: Some PrOperties of the Integral Bibliography iii Page '5 32 50 03 63 65 72 93 Chapter I: Preliminary Discussion Let (D,F,P) be a probability measure Space on which is defined a family of real valued random variables (r.v.'s) [X(t); teT) where T is a subset of the real line. We will always assume E(|X(t)|) .‘<'< 00 for every teT. Let [F(t); teT] be a family of sub a-fields of F with F(s)CZ F(t) .for every 5, tsT with s _<_ t. The family of r.v.'s {X(t); teT} is said to be well adapted to the family of sub a-fields {F(t); teT] if X(t) is F(t) measurable for every teT, and we will then write {X(t), F(t); teT] to indicate this relation. The family [X(t), F(t); teT} will be referred to as a stochastic process. After specifying a particular process (X(t), F(t); teT} we will often refer to it as the "X-process," in order to simplify writing. In many cases F(t) is the minimal a-field with reSpect to (w-r.t.) which the family of r.v.'s {X(s); SET, 5 5 t) is measurable. We will denote such a-fields by,’8 (x(s); seT, s 5 t). A process (X(t), F(t); teT) is called a martingale process if for every 5, teT with s S t, E(X(t)|F(s)) - MS) with probability one (a.s., a.e.), and is called a semi-martingale (super-martingale) process if E(X(t)|F(s)) 2 Ms) a-S- (E(X(t)lF(s)) S, X($) a-s-)- This can be restated as follows: Let "MW -fo(t,m)ap(m) for AeF(s) Then by the Radon-Nikodym theorem, there exists an F(s) measurable function which we denote by E(X(t)|l-'(s)) such that 2. us’tm) - AX(t,m)dP(m) - AE(X(t)|F(s))(m)dP(m) for AeF(s) Furthermore E(X(t)|F(s)) is unique except on an F(s) set of measure zero. The process {X(t), F(t); teT] is then a martingale if for every s,teT with s _<_ t, fx(s,m)dP(¢b) - fx(t,a>)dP(w) for AeF(s) A . A Correspondingly we can say the process is a semi-martingale if for every s,teT with s _<_ t, fX(s,o)dP(a>) S fx(t,¢p)dP(m) for every AeF(s) A A and the process is a super-martingale if for every 5, teT with s S t, fX(s,¢)dP(w) _>_ fX(t,a))dP(a)) for every AeF(s) A A He will assume from now on T is a closed interval, and hence it is no further restriction to assume T is the closed unit interval [0,l]. This will be assumed throughout the thesis. He can think of a process [x(t), F(t); teT] as a function X of two variables defined on the Space TXD. For each fixed teT, X(t,') is a r.v. defined on (D,F,P,) and is measurable w.r.t. F(t), and for each fixed men, X(-,m) is a real valued function with domain T. A sample function of the process is simply a member of the family (X(°,m); well} of real valued functions with domain T. We will be interested in analytic prOperties of the sample functions. However, in order to make probability statements about analytic properties of the sample functions, we must have separability of the process w.r.t. the class GPA of (finite or infinite) closed intervals. The process is said to be separable relative to CA if there is a denumerable subset T0 of T and a set.AeF with P(A) . 0 such that if As $ , and I is an open interval, then [X(t,u>)eA; teIflT]A [X(s,w)eA; teIflTo]t:A where [X(t,w)eA; teInT] - (w| X(t,w)eA; teInT}. In general we will let [-°-] denote the set of all men such that "---" is true. Separability w.r.t.), inf X(t,m) and HE X(s,w), 12."). X(s,co) teInT teIflT s€>t sé>t are all r.v.'s. \ He remark here that any process {X(t), F(t); teT} which is a.s. sample continuous is separable w.r.t. GA . By a.s. sample continuity we mean there exists a set.AeF with Pcn) - 0 such that if miA, than I lim X(s,w) a X(t,¢p) for every teT. sé>t Further, if T is a denumerable dense subset of T, then it is a O separating set. We proceed now to the definition of a quasi-martingale process. Definition l.l.l The process {X(t), F(t); teT) will be called a quasi-martingale process if there exists a martingale process [Xl(t), F(t); teT} and a process {X2(t), F(t); teT] with a.e. sample function of bounded variation on T such that F([X(t) = X,(t)+X2(t); teTl) = I When we say the process {x2(t), F(t); teT] has a.e. sample function of bounded variation on T we mean that except for aim» f with F(A) - 0, X2(°,w) is a real valued function of bounded variation over T. From now on we will always write [X]l or simply Xl for the martingale, and [X]2 or simply X2 for the process of bounded varia- tion in the decomposition of the quasi-martingale x. It is hoped that this will not be confused with the bracket notation used to indicate subsets of a. He now give some simple examples of such processes. Let [2(t), F(t); teT} be the Brownian motion process with T - [0,1]; i.e., the process has independent, normally distributed increments with E(Z(t)-z(s)) - O and E(|Z(t)-Z(s)|2) - Ozlt-Sl where a > 0 is fixed and s, teT. We assume 2(0)- 0 a.s. so that the process is a martingale process. we further assume F(t) =$(X(s); seT, s S t). Let X(t) - exp[Z(t)v] for every teT, where- v is an arbitrary positive real number. If u > O and t + u S l, E(X(t+u)|F(t)) - E(exp[Z(t+u)v]|F(t)) - £(exp[(Z(t) + Z(t+u) - Z(t))v]|F(t)) - exp[zv1 s - z>v1> .- X(t) exp[azvzu/Z] If we let t . 2 2 X2(t) -~/~ 02V X(s)dS for every teT, 0 r then the process (x2(t), F(t); teT) has a.e. sample function of bounded variation on T. That x2(t) is defined follows from the fact that the Brownian motion process is a.s. sample continuous. We show the process (X'(t) - X(t) - X2(t), F(t); teT] is a martingale. Again assume u > 0 and t + u _<_ I, then E(Xl(t+u)|F(t)) - E(X(t+u) - X2(t+u)|F(t)) - x exnlazvzu/Zl - :(x2(t+u)|r r]) . 0 r—>cn ogtgoo The problem of decomposing a process {X(t), F(t); teT) into the sum of a martingale process and a process having a.e. sample function of bounded variation on T parallels the above described decomposition of a super-martingale. For if we assume {X(t), F(t); tc[0,oo]] is a uniformly integrable super-martingale, we may as well assume we have the super-martingale [X(t), F(t); te[0,l]). Then if we have the above decomposition, the process having monotone non-decreasing sample functions has a.e. sample function of bounded variation on [0,l]. we will obtain necessary and sufficient conditions for a process {X(t), F(t); teT) to have the decomposition P([X(t) - Xl(t) + X2(t); teT]) . l where [Xl(t), F(t); teT] is an a.s. sample continuous martingale and the process {X2(t), F(t); teT} has a.e. sample function of bounded variation on T, and further if V0») denotes the total variation of X2(°,w) over T, E(V(u))) < oo- Chapter II Section I: Random stopping Let {X(t), F(t); teT} be a stochastic process defined on the probability space (fl,F,P). we will be interested in obtaining a sequence of processes {Xv(t), F(t); teT], v - l,2,..., where each process in the sequence has some specified property, such that F([xv(t) at M0 for some teT]) -—> o as v ->»oo. For example, we may want to define a sequence of processes (Xv(t), F(t); teT) v - l,2,... such that sup lxv(t,w)| < 00 for every v = l,2,... t,w and P([Xv(t) 4 X(t) for some teT]) —> 0 as v ->'ao. Such sequences are usually obtained by a random stopping of the process {X(t), F(t); teT). Therefore, we will consider briefly random stopping of a process (2, Loeve, pp- 530-535)- Let [x(t), F(t); teT] be a process defined on the probability space (fl,F,P) and let 1(00 be a r.v. defined on (fl,F,P) with range T. If for each teT, [1(w) S t]eF(t) (and hence him) < t}eF(t)), the r.v. The) is called a stapping time of the x-process. If the X-process is a.s. sample right or left continuous then we can define a new process [XT(t), F(t); teT} by randomly stOpping the X-process according to the stapping time f(a9. More precisely, if the X-process is a.s. sample right continuous, define mew) - new) c 5 .(w) I X(-r(w),w) t > 1(0)) and then using right continuity of the process and the fact that [7(w) _<_ t]-€F(t) it can be shown XT(t) is F(t) measurable for every teT. Actually, if one only requires [1(w) < t]eF(t), and defines XT(t,w) - X(t,w) t < 7(a)) 3 X(T(w),w) t 2 TD”) then XT(t) will be F(t) measurable if the X-process is either a.s. sample right or left continuous. The following is a standard theorem which we state here since it will be used extensively (2, Loeve, p. 533). Theorem 2.l.l If (X(t), F(t); teT] is an a.s. sample right continuous semi- martingale, (martingale) and if T is a stapping time of the process, then the stopped process [XT(t), F(t); teT} is also a semi- martingale (martingale). The next two theorems will also be used extensively in later work so we will prove them in some detail. Theorem 2.l.2 Let (X(t), F(t); teT} be an a.s. sample continuous process. There exists a sequence of processes [Xv(t), F(t); teT), v-0,l,2,..., each process in the sequence being a.s. sample equi-continuous, such that - P([xv(t) d X(t) for some teT]) < 2 V 10. Proof: By a process being a.s. sample equi-continuous we mean the following: There exists a set A with P(A) - 0, such that if e > 0 is given, there exists a 5 > 0 such that ' [X(t,w) - X(s,m)| < a when lt-sl < 5 for every wdA. (That A does not depend on teT follows from the fact that T - [0,l].) He now prove the theorem. Let (an; n _>_ 0] be a sequence of real numbers with e>e> ...>e>...>0 and lim 6- 0. For each n>0, let 0 l n n n -- (bnv’ v 2 0] be a sequence of real numbers with I 5n > 5nl>"'> 5nv>...> 0 and lim 5nv- O. The an s are arbitrary O and the anv's are to be chosen as follows: Because of the a.s. sample continuity of the x-process, for each en, n-0,l,... we can find a 5nv>0 such that ”(l 5UP |X(t.w) - X(s.w)| _>_ e 1) < 2‘0"”) It-slssm, n for each v - O,l,.... Let Tnv(m) be the first t such that sup |X(s,w) - X(s',w)| Z en . ls-s'len s,s' _<_t V If no such t exists we define Tnv(w) - I. Then for each n,v-0,l,..., O < Tnv(a>) _<_ l a.s. Tnv(w) is a stopping time of the process for every n,v-O,l,... since for any te(0,l] [Tm/(w) > t] a [ sup |X(s',a>) - X(s,w)| < en] |s~s'|<6 "" l1 s,s' S t V ll. Define, Tv(w) " 12f Tnv(‘”): V " o:l:"'° Then for each v - 0,l,... OSTV("°) S l. TV((D) will be a stapping time for the process if [Tv(w) 5 t]eF(t) for every teT. Actually it is sufficient to require that the set [Tv(w) S t] differ from an F(t) set by a set of measure zero. (I, Doob, p. 365) (As was mentioned, we could require only that [Tv(w) < t]€F(t) for every teT, which is obviously the case since 00 (no) < c] - {gonna < :1.) If te[o,l), then P([Tnv(0)) S t for infinitely many n]) to 3P T < I lim P 'r < ("00 "y” [ m(w) _ t1) new (“9:1th) _ t]) m. S lim 2' P([Tm v(w) m m-n n->m m-n Q 5 lim L P([ sup |X(t,w) - X(s, w)| > e m]) n->oo mm It- -sl_<_5mv 5 lim 2‘("+"")-o n->oo If Av(t) - [Tm/(w) E t for infinitely many n]. Then for thv(t), ”((0): t implies Tnv(w) S t for some n. We have P(Av(t))- 0 for every te[0,l) and every v - O,l,.... Letting ~Av(t) - £2 - Av(t), we have [Tv(w) _<_ t] = [121' TWO») 5 t] l2. - (tinf we») 5 no ~Av(t))U([1:f We») 5 tln Avm) l'l (D - ( u have») s :1) u ([inf we») 5 :1 0 Avon. n n-D CD so Int») _<_ t] - ("E-JI “MW 5 t1) =- [12f Tnv(w) 5 mm W” C Av(t). Hence [Tv(u)) _<_ t] differs from the F(t) set 00 U [1' (m) < t] by a set of measure zero. nv - m-O Define xv(t,w) a X(t,w) if t 5 who) - X(rv(w),m) if t > Tv(w)° Then for each v - 0,l,..., xv(t) is F(t) measurable for every teT and the process {xv(t), F(t); teT] is a.s. sample equi-continuous. For given any a >'O, if an 0 asv—>m. If in addition the X-process is such that lim r P([ sup |X(t,ua)| Z r]) - 0 r->o) t then E(|Xv(t) - X(t)|) —-> 0 as v —>o.'>for every tcT. If the x-process is uniformly integrable, sup E(|Xv(t) - X(t)|) —> O as v —> (D . t Proof: Define Tv(w) to be the first t such that sup |X(s,w)| 2 v. s < t If no such t exists, let 3(a)) - I. Then clearly 7v(a>) defines a stopping time for the process for every v - 0,l,.... Since the process is a.s. sample continuous on the closed interval [0,l], a.e. sample function has an absolute maximum. Define Xv(t,w) - X(t,w) if t S Tv(u>) - x(.v(m),a>> if t > We). Then for each v - 0,l,..., the process (Xv(t), F(t); teT] is uniformly bounded by v and is a.s. sample continuous. we have P([Xv(t) 1‘ KM for some teTJ) - P([Tv(w) < 1l) _<_ P([SUPIXU’MZ V1) t —->Oasv->a> Assume now that lim r P([sup lX(t,w)| Z r]) a 0. Then r->m t E(|Xv(t) - X(t)|) sf lxv(t) - X(t)]dP 4-] IXv(t) - X(t)|dP [13,092 t] [Tv(w)< t] lh. 'f |xv(t) -x(t)|dng va(t)|dP+ f |X(t)|dP [Tv(w) 0 as v ->’oo. 15. Section 2: Decomposition theorem. Let (X(t), F(t); tcT) be any real valued process with E(|X(t)|) < 00 for every teT. (2.2.l) Let (Trn, n 2:1} be a sequence of partitions of T. We denote the points of'rTL as follows: Assume ”IT H - max It . l- t .I —>O as n —>oo and let " 0<_j n,TTnC TTm and we will let TTnm,j -]Tmfl[tn’j, tn’jH] so that N n Trm - U flnm,j 1-0 The points OFT-Tnm j for j - 0,...,Nn will be denoted as follows ) fit <1: tn,j nmj,0 < tnmj,l nmj,k .+I" tn,j+l nmu He will always separate with a comma the variable subscript which denotes an arbitrary point of the partition and the subscripts indicating to which particular partition we are referrring. The variable subscripts will be written last. If l‘r is a partition of a closed interval B2,B]c: T with points a.aoU{largest element in TTn(s)] so that Tl’nm - lTn(s)U mm)- Then, for example, if 0 a a0 < aI < ...< av“: l T"! Z 2: end“) "- Z, Cn’jfl) for every n- l,2,.... 1-0 W n‘ai’ai-l-l) n We define {Xn(t),_Fn(t); teT), n . l,2,... as follows: Xn(t) - X(tn’j) if tn”. _<_ t < tnu’“ o _<_, 5 Nn-l-l. (2.2.2) - ' + . Fn(t) f F(tmj) if tn”. 5 t < tn’jfl o 5, 5 Nn l 17. We also define [in(t), Fn(t); teT], n - 1,2,... as follows: X2n(t) - Z Cmj (X) for every tcT. m(t) (2.2.3) Fn(t) - F(tn’j) if tn’j 5 t < tn’j.” o gj gun-l- l, or what is the same kzj X2n(t) - Z, cn,j(x) if tmk g t < tn’w o g k 5 Nn-l- l- 1'0 (2.2-11') If we let Xln(t) - xn(t) - x2n(t) for every teT, the process [Xln(t), Fn(t); teT) is a martingale. Clearly, the process {X2n(t), Fn(t); teT) has a.e. sample function of bounded variation on T since a.e. sample function takes on only a finite number of distinct values. Then each process (xn(t), Fn(t); teT] is a quasi- martingale. If we assume the X-process is continuous in the mean, then it is the limit in the mean of a sequence of quasi-martingales, [Xn(t), Fn(t); teT], n 2;l. Further, if for each teT, the sequence (X2n(t); n 2;!) converges in the mean to a r.v. X2(t), then the process {Xl(t) - X(t) - X2(t), F(t); teT] is a martingale. For any teT, let kn(t) denote the last k such that tn 5 t. k ) Let s,teT, s g t, and let AeF(s). Then LXIHMP- AxmdP- fAXZMdP l8. . AX(t)dP- lim (jAl (Z c Hxndwf (Z CnJ-(Xlldl’) n-> co 77""(5) 7T" (5, t) .- fo(t)dP - fA X2(S)dP -n_l_;mm A( Z cn’j(X))dP- 7771(5) 13) Now f( L Cn’j(x))dPI fa): c .(X)|F(s))dP A 7mm) 7r 0 as n ->’oo. So we have AXI(t)dPI '[x X(t)dP- AX2(s)dP- AX(t)dP+ fa X(s)dP I ‘X; Xl(s)dP . This however still tells us nothing about the process {X2(t), F(t); teT) even when it exists. we now look for sufficient conditions for the Xz-process, when it exists, to have a.e. sample function of bounded variation on T. 19. The following lemma, though trivial, is a key starting point. Lemma 2.2.5 If {1TB; n 21'} is defined as in 2.2.], then for any process (X(t). F(t); teT} HZ lcmlxlll Tl' n is monotone non-decreasing in n. Proof: Let m > n. Then 77'“:le and we can write _""l "'l EQ. ”@100“ " “Z24 Icnmj,k(x)l) TTm TTnTThmj 55% “A. lcnmj,k(x)| anj” _>_:(Z' 1:424 cnmj,k(X))anJ.)l) «(Zlcmlxm fin finmj fin we make the following observation: Let (X(t), F(t); teT) be a quasi-martingale with [X]]I XI and [X] I X . If a.e. sample function of the x process is continuous, 2 2 2 then V I li As . (w) .30. ll m(lel 77n is the total variation of X2(-,w) over T for a.e. w, and it is a random variable. Assume E(V(w)) < 00, then “X lend-00h =- eQJ Icmllel) 77.. ”n 200 5 NZ lAn,j(X2)l) 3. NW») < oo. n and hence lim HZ IC .(X)|) < oo . n->oo "’J n In fact, if"TT is any partition of T, there exists K independent x) of 7T, such that 5% |c1(X)|) 5 xx < 00. In view of this and Lemma 2.2.5 we restrict out attention to those processes satisfying the following condition. (2.2.6) There is a sequence of partitions UT"; n 2 l] of T, as defined in 2.2.] such that lim E(Z|cn’j(x)|) 5kx_ I, be defined as in 2.2.3. If the process [X2(t), F(t); teT} is such that P([lim X2n(t) I X2(t); teT]) I l n then the X -process has a.e. sample function of bounded variation 2 over T. Furthermore, if the X -process is a.s. sample continuous, 2 the total variation V(w) of X2(-,a>) over T is a r.v. and 2|. E(V(u>)) _<_ "gnoouz' |cn’j (x) |) _<_ “x n P roof: By2..,26 Kx_>_ lim “2|an .(Xll) n->m 77?. >5 li c x , h f l - l , _ ("shim 2|" j( )I) so t at i we et Kx(w) "T32, 24 Icn,j (X)| nn Trn then Kx(a>) is a.s. finite and integrable. If 0 n a < a < ...< a l. l is any partition of T, then 0 l k-l- 33. Z |A1(x2)| - :MI um (Z cjooll 1'0 am TT n(ai’ai+l) < _l___1m Z 2, |cn J.m:_>_(x)|-- main" j(X)|IKx(w))I lim ZlAn j(X2)| a.s.) n7T->oo 7Tn By our previous remarks and Lemma 2.2.7, if the process (X(t), F(t); teT} is continuous in the mean, satisfies condition 2.2.6 and if {X2n(t), Fn(t); teT}, n 2 1, defined as in 2.2.3, is 22. such that there exists a process (X2(t), F(t); teT] with P([ lim X2n(t) I X2(t); teT]) I l n-D’oo and also E(|X2n(t) - X2(t)|) —> 0 as n —> co for every teT, the X-process will be a quasi-martingale with [XlzI X2. These are the basic conditions we will seek to satisfy in obtaining first sufficient conditions for a process (X(t), F(t); teT] to be a quasi-martingale. we will need the following theorem. Theorem 2.2.8 Let (X(t), F(t); teT] be a second order process. Let D2,B] be a closed sub interval of T and let a I ao < al <...< an-l-l' B be a partition of [a,B]. Let c >’0 be given. If e Isup'[ max X(Bw)-X(a,w)]' I .Am [mmzxnlgok )l e] Uo Now s(|é~c(x)|)-Z fIchm». écflundp. m-O An1 k80 n For each m- 0,l,...,n, we let AmI ick“), BmI Z Ck(X), and kIO kIm+l replacing A.and F with.Am and F6 and B with ea 5’ the above argument ) gives "11 tug-Joke (x)| 2) > ":0 jam (6- «(1,5) 2dr m = (e- ea 52) P([...”é’: lgockun > all and hence the theorem is true. Corollary 2.2.9: Let (X(t), F(t); teT} be a second order process which is a.s. sample equi-continuous. Let {TTn; n 2:1) be a sequence of partitions of T as defined in 2.2.l. Given any a >s0, there exists an n(e) such 24. that if n 2 n(e) then for every m 2 n, k P( ma max C . (X) > ]) [OSJXSNn OSkSknmJ l12:0 "WA |— e 5 “LIZ. enmj’kmlzll («s-en)2 ”h nhmj where e I sup' max sup |X(t,u>) - X(S,(D)I < e CD 051 gun tn’sz,t_<_ tn.J‘+| (sup' denotes supremum over the set of equi-continuity). a) Proof: Because of the a.s. uniform sanple equi-continuity of the X-process, given any 5 > 0 there exists 5(6) such that sup' sup |X(t,w) - X(s,a.l)| < e . a: |t-s|§5(e) Let n(c) be such that ||fl;1||< 5(6) if n 2 n(e). Then an I sup' max sup [X(t) - X(s)| < e to 0994" tnj<_s,t$tnj+l and for all m _>_ n, sup' max max |X(t . )-X(t .)| w 061) " 0'0, there exists n(e,n) and 5(c,n) such that if n 2 n(€.ll) P([ sup IXn(t) - Xn(s)|>'e]) < q. |t-s|55(c,n) Then there exists a subsequence of processes {Xn (t); tcT) k I l,2,... and a process (X(t); teT] such that a) P([lim xn (t) a X(t); teT]) = l k k and b) The X-process is a.s. sample continuous. Proof: We first show that conditions i) and ii) imply TIE’ P([sup |Xn(t) - Xm(t)| >.€]) I 0 for every 6 >.0. nyrv€>oo t Let (TV; v 2:1] be a sequence of partitions of T with CD TICTZC and U TV I To. Let e,n>0 be given. First vIl choose "](€:D): 5(e,n) such that P([ sup H£X (t)-Xn (s)|>a§ ]) |t- s| |<:5( e, :1 <3 26. for every n 2. n](6,n). This can be done by condition ii). Now choose v such that max |t tv jl < 5(6,n). Next, choose ° ) v,j+l- nv(6,n) such that P([max lxnltm) - Xm(tv pl > §-]) < 11/3 J J for every m > n _>_ nv(€,1]). This is possible because there are only a finite number of points in TV and for each tv,j€Tv’ Pnl-inho xn(tv,j) exists. Now let n(6,n) I max (nl(6,q),lnv(6,n)} and consider P([sgp lxnlt) - xm(t)| > e]) I P([max sup IXn(t) - Xm(t)| > 6]) J tvajs t '<" tvyj'l'l _<_ P([max sup IX (t) - X (tv .)| > .3/31) 1 ‘m'S-‘S ‘v.j+l n n ’J 4‘ P([max lxnuv j) " xm(tv jll > 6/31) J 2 ) + P([max sup Ixm(tv,j) - Xm(t)| > e/3]) < n J ‘m‘st 5 ‘m‘u Now lim P([sup IX (t) - X (t)| > 6]) 0 for every 6 > 0 implies n m n,m—>oo t there exists a subsequence (Xn (t); t6T], k 2;l, a process k (X(t); tcT] and a set A with P(A) :- 0 such that if mm then lim (sup IX” (t) - X(t)|) I 0. k->co t k we have now established a). we now proceed to show the process (X(t); t6T] is a.s. sample continuous. Let XI;(t) I Xn (t) for every teT and k I l,2,.... k 27. For every n = l,2,..., by condition ii), we can find a kn and a 5(n) such that for k 2 kn P([ sup lx'(t) - X'(s)| > n"]) < 2‘n . i k k It-SIS o(n) * a Let Xn(t) - Xk (t) for every teT; n a l,2,.... Let n * * 'I' - An - [ sup IXn(t) - Xn(s)| > n '1 lt-SIS 5(n) oo *- a) * x- __..-x- -ll- *- IfB-UA,thenA-limA -UBandP(A)=0. k n n k n-k k-l * If “8", then for every n _>_ k, * «u- -1 sup IXn(t) - xn(s)| g k . |t-slgb(k) at- Now if (MAUA , and if e > O is given, sup |X(t) - X(s)| S 2 sup Ix:(t) - X(t)| + sup Ix:(t) - x:(s)|. lt-sl 55 t |t-s| _<_?» at First choose ko(w) such that 2 sup |Xk(t) - X(t)] < e/Z for every t k 2 k0(d>). This can be done since MA- Next choose kl(cu) such that «13* and k (w)" < e/Z. Then if k(a)) - max [k (w), k (03)] and kl(w) l O l 5 S 5(k'(w)), we have sup [X(t) - X(s)| S e lt-sl s. 5 Then for mu A*, FUN) A*) . 0 lim sup [X(t) - X(s)| a 0 . 5—> o |t-s| 55 28. Theorem 2.2.ll Let (X(t), F(t); teT] be a uniformly bounded a.s. sample equi- continuous process satisfying condition 2.2.6. Then the process is a quasi-martingale with [x] - x where X (t) - P lim x (t) for 2 2 2 n->oo 2n every teT, and the processes (X2n(t), Fn(t); teT] n - l,2,... are as defined in 2.2.3. Further, the X ~process is a.s. sample 2 continuous and if V(a)) denotes the variation of X2(-,m) over 1' then E(V(¢u)) < co . Proof: Because of Theorem 2.2.l0, Lama 2.2.7, and our previous remarks we need to show the following: to i) P lim X2 (t) exists for each te U N - N. n n n9 oo ".1 ii) Given 6, 'q > 0, there exists n(e,n) and Ham) such that if n 2 n(e,n) P( X(t)-X(s)> )< [It-siugbkm) l 2" 2n I e Tl and iii) For each teT, the sequence {X2n(t); n _>_ l) is uniformly integrable in n. we first show 111) is satisfied by showing E(|X2n(t)|2) g K< 00 for every n _>_ I and teT- We have e<|x2n(t)|2) = all cmjmlz) 17am . HZ. |cn’j(x)|2 + ZZ cn’jmtz‘ cn’k(x))) TT (t) m(t) k>J 29. - HZ Icmmlz) + ml cmmuz cn,k(x>lrnj)) TT (t) m(t) k>J 5 “Z lcn,j(x)|lAn,j(x)l) + 22(24 ICn’j(X)||X(tn’j(t))-X(tn’j+l)l) TTn(t) m(t) where j(t) is the last j such that tnj _<_ t. ) Then 2 ‘-l E(|X2n(t)| ) 5&1qu |cn,j(x)|) g 6MxKx m(t) where H - sup|X(t (0)] and K 2 lim HZIC .(X)|). X t,w ’ X n->m ""1 TTn He now prove i) by showing E(|X2n(t) - X2m(t)|2) --> 0 as n,m —> 00 for every teTT - In so doing we will pick up an inequality which will allow us to prove ii) inmediately. If ten, then there exists nt such that te W" for every n 2 "t' We assume now that m > n 2 nt. Then x2n(t) ‘- Z “Z cnmj,k(x)anj) m(t) mm] and x2m(t) " Z 77.. Let Vnmj (X) a Z cnmj,k(x)' 77'nmj Then e<|x2m(t)- xanIZ) an); (v rrnm 2 nmj- 5(Vnmj(">|FnJ-))| ) 30. 2 . mi | vnmJ.(x) - la(vn mJ.(x)|n=m)| ) fl;&) .. 5(24 lvnmjo‘ )I 2) - 2Q: |E(vnmj(X)|Fn’J)|z) Ugh) //n(d < EQ‘ IvnmJ( )lz) 77} . all |cnmj kW) Trn 777w ,- ‘ + 25(21'211 cnmj’k(x)E(2L‘ Cnmj,i(x)anmfi,k)) mflnmj i>k E A . (X) 9.1 “1 (X S. (OQMQNn ngiknmjl nmJ,k '12?le cnmJ k )l) m] + 2E(o)|, then " w 0_ n. Hence E(|X2m(t) - X2n(t)|2) 5 E(Z|ZJC k2(X)| ) _<_ 36" Kx ——> o as cnmj, 7Tn mm m > n -> oo . Hence 1) is proved. To prove ii), assume 6 < min ltn,j+l' tn,_j| and then for OSJSNn every m 2 n, 3|. P([ltfgfsslxmm - XZm(S)l > 36]) k _<_P([ max max Ichm' 1(x)| > 6]) O4 Cnmj’km|2)/(«s-<-:n)2 1T.n Trnmj 2 5 3enKx/(e-en) -—> 0 as n —> m Hence if e,n are given, we can choose n(e,n) such that BenKx/(e-en)2 < 1]- If we then fix n > n(e,n) and choose 5(e,n) < "11;”, then for every m 2 n, P x (t -x ( >3 < . ([ltEETSMemll 2'“ ) 2mm 6]) " Hence condition ii) is satisfied and the theorem is proved. 32. Section 3: Main decomposition theorem: In order to prove the main decomposition theorem we need to do more preliminary work. He now investigate the uniqueness of the decomposition of a quasi-martingale process. we need the following lemma. Lemma 2.3.] Let [Y(t), F(t); teT] be a martingale process having a.e. sample function continuous and of bounded variation on T. Then P([Y(t) - Y(0); teT]) . 1. Proof: Since the Y-process has a.e. sample function continuous and of bounded variation on T, if V(t,w) denotes the variation of Y(-,a)) over [0,t] then V(°,a9 is continuous and monotone non-decreasing on T for a.e. a» Further V(t,-) is a random variable measurable w.r.t. F(t) for every teT. As in Theorem 2.1.3, we define ¢v(t) to be the first t such that sup |V(s,w)| RV or sup |Y(s,w)| 2v. 5 SLt 5 fit If no such t exists let Tv(u)) =- l. Clearly, TV((D) is a stopping time of both the processes {Y(t), F(t); teT) and {V(tfib), F(t); teT}. Define Yv(t,w) a Y(t,a)) if t _<_ who) a Y(Tv(w),0)) if t > Yvon). By Theorem 2.l.l, for each v a l,2,... the process {Yv(t), F(t); teT} is a martingale. Furthermore, for each v - l,2,... the Yv-process has a.e. sample function continuous and of bounded variation on T. 33- As in Theorem 2.l.3, for a.e. u) , there exists v(m) such that Y(t,(D) - Yv(t,u>) for every teT if v 2 v(w). It is also clear that if vv(t,w) denotes the variation of Yv(~,w) over [0,t], then Vv(t,w) - V(t,a>) if t S Tv(w) " V(Tv(w):w) if t) wk”)- This follows simply from the fact that for all s _<_ t S 1v(w), Y(s,a>) - Yv(s,w), and for t 2 ¢v(a>), Yv(t,a>) is constant. Also we have sup Ivv(t,w)| _<_v , sup [Yv(t,u>)| 5v t,w t,a> We now show that for every 1/ - l,2,... P([Yv(t) - Yv(0); teTl) = I. Let UT"; n 2 I] be a sequence of partitions of T as defined in 2.2.l, co and let 1T- U IT . Let ten. Then there exists n such that nal n t te TTn for every n _>_ nt' Assume n 2 nt. Now ulvvm - vv‘x)l n’J( Vll V( . )) Therefore P([YV(t) - vv(0); teTTJ) = I- Since 17. is dense in T and Yv is a.s. sample continuous, we have P([Yv(t) - Yv(0); teTl) - I. and this is true for every v :- l,2,... . Since for a.e. a), when v is sufficiently large Yv(t,w) - Y(t,w) for every teT, it follows that P([Y(t) - Y(0); teTl) = 1 Theorem 2.3.2 If (X(t), F(t); teT] is a quasi-martingale with the following decompositions P([X(t) - X,(t) + x2(t); teTJ) - I P([X(t) - XT(t) + X;(t); teT]) a n where the x1 and XI, 1 a 1,2, are a.s. sample continuous, then P([x, - x’,‘),w) if t > Tv(u>) . Then for each v a 0,l,... the Xv-process is a.s. sample equi- continuous and uniformly bounded by v. we have P([-rv(ao) < l]) g P([T;(LD) < l]) + P([1-3(w) < l]) —> O as v -§'oo. Recall that [Tv(w) < I] = [Xv(t) + X(t) for some teT]. 36. Then there exists a set A with P(A) - 0 such that for (MA, there exists v(u>) such that if v 2 v(a>) X(t,w) :- Xv(t,u>) for every teT. If r P([sup [X(t)] 2 r]) —> O as r -—>oo , then t E(|Xv(t) - X(t)|) ——> 0 for every teT, for E(|Xv(t) - X(t)!) . f m(t) - X(t)]dP + f |x,(t) - X(t)]dP [., as v —>oo and X(t) is integrable. The first term is bounded by vP([1-‘;(w) < l]) + VP([T;"((D) < l]) 5 v2"’ + vP([T;(w) < l]) S v2-V + vP([sup |X(t)| 2 v]) —> O as v —>oo . t We now prove another lemma which will lead us to the main theorem. Lemma 2.3.h. Assume {X(t), F(t); teT] is a.s. sample continuous and is such that rP([SUp |X(t)| 2 r]) —> o as r -—> oo. t Let the sequence of processes [Xv(t), F(t); teT), v 2 0, be as defined 37- in 2.3.3. If the X-process satisfies condition 2.2.6, then each Xv-process also satisfies condition 2.2.6 and the bound K is inde- pendent of v. Proof: We assume lim E(Z|CnJ .(X)|) co n every v 2’0. LetT n-Hzlcmj (x)|) and HT - J.V.(x)|) First 77 n consider Siam, 2f (ICn’J(XV)|' lcn’j(X)l)dP<}—' fie) ,>j(xv') cn jun“, [TV 7Tn [3,0102 taxi] Trn j] 52‘ f lxvum“) - xun’jnep _<_ (|x(tn,,-..)| + v) d? Tr" [Tv(m)_>_ tn,j] 7T" [tn’js 13,0”) |>dm “n, ks- 13(0)) _thJ.] - L (f ICnJJ(X)|dP 4-] |cJJJJ(X)|dP) "'n t.,,(w> |dn 7Tnl (w): tn J] 77'. [new th J] -2 flc JJJJJ(X)|dP 7Tn [7(wn)_thJ1 77.. [.,) oo, both vP([1-v(w) <11) and f |X(l)|dP go to zero. [TJ(“W<'] Hence if KJJ :- lim HZ ICnJJ(XJJ)|), then Kv ——> Kx as v —-> a). w€>ao n Main Theorem 2.3.5. In order that the a.s. sample continuous, first order process (X(t), F(t); teT) have a decomposition into the sum of two processes P([X(t) . X,(t) + x2(t); teTJ) = I. where {X‘(t), F(t); teT) is an a.s. sample continuous martingale process and the process [X2(t), F(t); teT] has a.e. sample function continuous and of bounded variation on T with E(V(a>)) < a), where V(w) is the total variation of X2(-,u>) over T, it is necessary and sufficient that hl. 1) lim rP([sup |X(t,w)| 2 r]) a 0, and n€>oo t ii) Fer any sequence of partitions {TT;, n 2;l] of T with lllTnll -J;-> o and ITJc 772: lim “2ch .(x)|)5k oo n’J X n Kx being independent of the sequence of partitions. Proof: we first prove the necessity. If (X(t), F(t); teT) is a quasi-martingale with the stated decomposition, we have already indicated that ii) is true. we need then to prove 1). Consider rP([sup |X(t.w)|2 r]) rP([SUP IX(t.w)- X(0.w)+ X(0.w)| _>_ r]) t t g rP([sup |X(t,w)- X(0,w)| Z r/2]) + rP([|X(0,w)| _>_ r/2]). t Now rP([IX(0;w)l Z r/2]) S 2] ”(0’0”)“? -—> 0 as r -> CD (mm) l2 r/Zl Consider then rP([sthp IX(t,w) - X(o,w)| 2 r12]) = rP([stJap l(xl(t,w) - x,(o,w)) + (x2(t,u>) - x2(o,a>))| a rim 5 rP([SLJIP IX,(t,w) - X.(0.w)| Z I‘M) + rP([stJap Ix2(t,a>) - x2(o,w)| a r/llll- Now {(Xl(t) - XJ(O)), F(t); teT} is a martingale and hence by Theorem 3.2, sec ll, Chapter VII of Doob, we have #2. P([s:p lxl(t,w) - x,(0.w)| _>, r/uJ) su/r E(|x,(l) - xl(0)|) -> O as r -—> a). By the same theorem rP([sup |xJ(t,w) - xJ(o,m)| 3 rm) _<_uf |xJ(I) - xJ(o)|dP -—> o t [sup |xl (t)-xl (o) |_>_ :14] t as r ->’oo. Consider now rP([sup |X2(t)- X2(0)| 2 I’ll-l]) S rP([sup |V(t,w)| 2 r/hJ) t t S rP([V(l,a)) 2 r/h]) < hf V(l,(.l))dP —> 0 as r —> a), [V(l,u))2 r/l-l] where V(t,w) denotes the variation of X2(-,w) over the interval [0,t]. Hence, if the quasi-martingale (X(t), F(t); teT] has the decomposition stated in the theorem, conditions i) and ii) are satisfied. we now prove the sufficiency of i) and ii). Let the sequence of processes [Xv(t), F(t); teT] v - l,2,... be defined as in 2.3.3. Then P([Xv(t) a X(t); teT]) ->'l as v ->'oo. By assumption 1), E(|Xv(t) - X(t)|) ->’O as v -+>'oo for every teT. By Lemma 2.3.h, each process [Xv(t), F(t); teT} v a l,2,... satisfies condition 2.2.6, the bound K being independent of v. Then by Theorem 2.2.ll, each process has the decomposition P([Xv(t) = le(t) + X2v(t); teT]) a l where [le(t), F(t); teT} is an a.s. sample continuous martingale process, and the process [X2v(t), F(t); teT] has a.e. sample function of bounded variation on T. Further h3- X2v(t)=-P lim > (In __. JJ(XJJ) for every teT I and so if Vv(t,w) denotes the variation of X2v(-,w) over [0,t] we know by Lenlna 2.3.4 and Lemma 2.2.7 E(Vv(t,a>)) 5 ngnJJJJ 5(J2JJ |cJJJJ. (xv) |) _<_ x < as n for every teT, v . l,2,.... *- It is clear that 11(0)) 5 72(40) 5 a.s. Let v > v and let * Xw(t,w) a le*(t,w) if t S. Tv((D) le*(Tv(‘”):‘°) if t > Wk”) * X2v(t,w) X2v*(t,w) if t S Tv(w) X2v*(7v(w),w) if t > WW . a «it By Theorem 2.2.], xlv is a martingale, and clearly the XZV-process *- has a.e. sample function of bounded variation on T. Further xlv and * x2v are a.s. sample continuous. Since Xv(t,u>) - Xv*(t,w) if t 5 Tv(w) a Xv-x-(Tv(w),:) if t > Tv(w) we have P([xv(t) = xh,(t) + x2v(t); teTl) = I P([xv(t) =- xTvm + x:v(t); teT]) .. l . "-I Now X2v(0) = P lim 2.. CnJJ.(XJJ) = 0 for every v a 0,1,... so n->oo .-.- un(o> to. that X2v(0) . x2v*(°) - 0 a.s. And hence by Theorem 2.3.2, we have if P([x2v(t) - x2v(t); teTl) - I For a.e. u), i.e., except for weA, where P(A) - 0 there exists a v(w) such that for all v 2 v(a)), rv(a>) - I. Then for v* > v 2 v(u>) X1V*(t,w) - X1v(t,a>) for every teT. (i :- l,2) we define, for (MA, X (t,a>) - lim X (t,w) for every teT 2 2v v-> oo Xl(t,w) - lim le(t,w) for every teT. v-> Q) And hence for every mm there exists v(a)) such that if v 2 v(w) X1(t,a)) - Xiv(t,w) for every teT. (i a l,2) Now, for every v a 0,l,2,... and (um X1v(t,a>) =- Xi(t,w) if t g Tv((D) 1 a 1,2 I- Xi('rv(w),w) if t > Tv(a)) ( ) If V(t,w) denotes the variation of X2(-,w) over [0,t] , V(t,u>) is finite since there exists v(a>) such that X2v(t,w) a X2(t,w) for all teT and hence V(t,w) . Vv(t,cu) for all teT. Clearly Vv(t,w) a V(t,w) if t _<_ who) = V(1-v(w),w) if t > 'rv(w) and P([ lim Vv(t,a>) a V(t,w); t€T]) a l. v->m 1.5. Since Tv(w) is a.s. non-decreasing in v, Vv(l,w) :- V(Tv(m),w) is monotone non-decreasing in v. But lim E(Vv(l,w)) _<_ K, v-D'oo and hence by the monotone convergence theorem lim E(Vv(l,w)) - E(V(l,w)). v-D'oo Now |X2v(t,w) - X2(t,w)| = |X2(t,w) - X2(t,w)| if t _<_ 1v(tu) :3 |X2(t,w) - X2(‘rv(w),w)| if t > Tv(w). So sup |X2(t,w) " X2v(t,w)l " 5UP IX2(t,w) " x2(Tv(‘°))‘-D)l t t> TV a) ,<_ sup (V(t;‘°) " V(TV(CD),CD)) _<_ V(‘:‘”) ' Vv(l,w). t>Tv((D) And hence E(sup |X2(t) - X2v(t)|) S E(V(l,w) - Vv(l,w)) —> O as v -—> to . t Now since E(IXv(t) - X(t)|) —> o and E(|X2v(t) - X2(t)|) —> o as v —> oo for every t€T. we have E(Ile(t) - xJ(t)|) —> o as v -—>'oo and hence Xl being the limit in the mean of a sequence of martingales is itself a martingale. we now make a few remarks concerning the decomposition in Theorem 2.3.5. First, the process (X(t), F(t); teT} is uniformly integrable in t since the process [Xl(t), F(t); teT} being a martingale process closed on the right is uniformly integrable, and {X2(t), F(t); teT) is uniformly integrable because it is dominated by V(l,a9. l+6. Secondly, having already proved the decomposition we can easily show .——. X (t) - P lim 2. C (X) for every teT. 2 n-> oo n,j TTnlt) Let (Xv(t), F(t); teT) v . 0,l,... be as defined in the theorem, so that for each v - 0,l,... Xv is, by Theorem 2.2.ll, a quasi- martingale. Let P(I[Xv(t) - le(t) + X2V(t); tch) - l where [x91]. xlv and [Xv]2- X2v. In Theorem 2.2.1] we showed P([SUP IZJ Gnu-(Xv) - X2v(t)| > cl) -—> o t m(t) as n —> on for every 6 > 0. In Theorem 2.3.5 we showed P([sup |X2v(t) - X2(t)| > 6]) -> 0 t as v —>m for every 6 > 0. we will now show P([sup [Z (cmj (X) - cmj (Xv))| > 6]) —> 0 " Wt) as v -> a) uniformly .111 2. Let c > 0 be given. Let x[ 1‘ denote the characteristic function of the set [...]. Then P([sup l2. (anJ.(x) - anJvam >e1) - ”D‘twmm 52" l}. (anJ.(x) - anJ.(xv))|><-:n + P(lX[.J(a)..u SJ... IJZJJJJJanJM) - Cn,3(".))|>€1) ’ n 47. Now the first term on the right is bounded by P([rv(0>) < l]) which we know goes to zero as v --> m. The second term is bounded by r'": P([X[Tv(m)_]] 12f chJJJ.(x2) - cJJJJ(x2v)| >cl) n E 2.. f Icn,j(x2) ‘ cn,j(x2v)|dP Trn [gm-I] IA I -fi 5- 2 2. f “'An 1(le ‘An.J‘"2v’|'Fn.J’dP TI [TV (09> t n,j] I -—l n E- g! f |X2(thJ+l) ' X2v(thJ+J)|dP . > . n [Tv(m)_ thJJ] , so that But X ) unless 7v (to) < t 2(tn,j+l) " "‘21" n, j+l 2.. f lx2(tn,j+l) ' x2v(tn,j+l)ldp 77.. [.,(w)z th J1 n,Jj-l-l au— 1 " E Z f l"2(tn ,j-I-l) x2v(tn,j+l)|dP Tr" [t nJ-S TV (m)< tn :J'H] ((32. 2V(l,m)dP =- é-f 2v(l,m)dP 7Tn [th J._<_ T JJ(o) 00 since P([-rye») O as v —> co, and v(l,w) is integrable. 48. Now if e >'O is given, P([sup IZJ anJm - x,(t>|>e1) 59([sup IL (anJm - anJ.(xv))| > e131) ‘ m(t) + P([sup IZ “..,j‘XvI - x2v(t)| > e/31) ‘ mm + P([sup Ixzvm - x,(t)| > em)- t Given any n >‘0, we can first choose v such that the first and third terms on the right are less than n13 for every n - l,2,.... For this fixed v we can make the second term less than n13 by choosing n sufficiently large. An immediate corollary to the main theorem is the following. Corollary 2.3.6 If {X(t), F(t); teT) is an a.s. sample continuous semi-martingale, then it has the decomposition stated in Theorem 2.3.5 if and only if 1) lim rP([sup |X(t,w)| _>_ r]) =- o r—D’oo t In particular, if the X-process has a.e. sample function non-negative, then i) is always satisfied. Proof: we need only show condition ii) is satisfied. If (TTn; n 21l} is a sequence of partitions of T as defined in 2.2.l, then 149. eQJ lanJ.(x}|) ”()4 |E(AnJJ.(X)anJJ)|) 77.. Tl'n - sQJ |e(x(thJ,,)|rnJJ) - X(thJHI 7’7. 'fi - “Z (e 0 as r —>oo , [sup X(t,w)2 r] t and therefore i is satisfied. If we recall the example given in Chapter I, where X :3 exp[Zv] v > 0, 2 being the Brownian motion process on [0,1], we see that the X-process is an a.s. sample continuous non-negative semi-martingale since exp[tv] is a continuous, convex and non-negative function. The corollary tells us the X-process is a quasi-martingale. we note that when the X-process is a semi-martingale, our conditions coincide with those given by Johnson and Helms. 50. Section 4: Particular results: In this section we concern ourselves with some Special theorems and results which will be used extensively in Chapter III. we first prove a lemma which will be quite useful and which is really just an observation. Lemma 2.4.1 Let (X1(t), F(t); teT], i - l,2,...,k be arbitrary processes. Let {TTn, n 2;l) be a sequence of partitions of T as defined in 2.2.]. For each n - l,2,... let fn(-) denote a Baire function of k(Nn+ 2) real variables. Assume that for each i - l,2,...,k, {X1v(t), F(t); teT), v - l,2,... is a sequence of processes such that P([X1v(t) X1(t); teT, l g i S k]) -—>l as v -D’ao. If for each fixed v a l,2,..., P lim f (X n );l§_is_k,0_<_j_<_NJJ-rl) n->cn iv(tn,j aPlim f- a? exists, rr€>oo nv v thenPlim th .;l _ _, _J_ n > - P lim f. a f. exists. n->oo n Moreover, P lim f. exists and is f'. ‘v€>oo Proof: we observe that ?hv converges to f; in probability uniformly in n as v -§'oo. For sup P([]?‘JJJJ- Tnl > 0]) l - inf P([l'f'mJ- fn| 5 0]) n n 51. e1 - 12f P([xiv(thJ) a xi(thJ.); I 51 < k, 0 SJ 5 Nn+ l]) _<_] - inf P([Xiv(t) = X1(t); teT, I 5 i _<_ k]) n =l-P([X )aX1(t);teT,l§_i§k]) -->0as v—>tn. iv"t Then pm?“- in] > .1) _<_ P([IFJJ- Invl > 01) + P([|Tnv- va| > 5]) + P([lfmv- Tm] > 0]) where e >'0 is arbitrary. Let 5 >'0 be given. We first choose v such that the first and third terms on the right are less than 5/3, then for that fixed v we can choose n and m to make the second term on the right less than 5/3 because P lim f; a f; exists for every n-arao v - l,2,.... Hence P lim T = f exists. n n-D'oo To show P lim f' exists and is f, consider v€>oo P([I-f-v- ?I > 6]) S P([l-f-v- "I’m" > (5/31) 4- P([]fnv- fn| > GB” + P([]fn- f] > e/3]) where e >.0 is arbitrary. Let 5 >.0 be given. we first choose v such that the second term on the right is less than 5/3 for every n, and for that fixed v we can choose n such that the first and third terms on the right are less than 5/3. The lemma is now complete. with this lemma it is easy to prove a useful theorem concerning the decomposition of a particular type of semi-martingale. Let (X(t), F(t); teT) be an a.s. sample continuous second order 52. martingale process and let é - X2. Then the process [§(t), F(t); teT] is an a.s. sample continuous, positive semi-martingale and by Corollary 2.3.6, g is a quasi-martingale. Let [E]l - 5| and [t]2 - {2. Then we know :2 has a.e. sample function continuous and of bounded variation over 1'. Further, if V(a>) denotes the total variation of §2(-,u)) over T, E(V(w)) < co, and '"‘l §2(t) - P lim 2:. C .(E) for every teT. n-> (D n, J rrnm Theorem 2.4.2 If (§2(t), F(t); teT] is the process defined above, then 52“) - P lim 2 [AJJJj (X)]2 for every teT. ”wnnm Proof: J we know §2(t) . P lim 2. cm] (s) "Q °° m(t) - 9 lim 2 cIn (x2) - 9 lim Z E([A (x)]2|r .) . a] .., naj n:J n-> (”77.n(t) n > mTTn(t) Assume first that the X-process is a.s. sample equi-continuous and uniformly bounded. Then all ([AnJJ.(x)12 - E([AnJJ(x)12|FnJJ))|2> m(t) - “Z (IAJJJmW - IsuAnJJIx)12|rnJJ)|z>> ‘ rrn(t) 53. SHZI JJ(x)| “) _<_:(JJ max< JAMWZ}: IAnJJ(x)|2) Trump” mm 5 enE(ZJ IAnJJ.(X)|2) g ant-:(Ixm - X(OIIZI. m(t) where en . sup max [An .(X) [2 --> 0 as n -> a), because of the w o gjgnn ” a.s. uniform sample equi-continuity. Now assume the X-process is an a.s. sample continuous second order martingale process. Let (Xv(t), F(t); teT} be as defined in 2.3.3. Then by Theorem 2.l.l, each Xv is a uniformly bounded a.s. sample equi- continuous martingale process. For each v a 0,l,..., if 5V - X3 , then tv has the decomposition gv - glv + g2v and by what we have just shown 2JJ(t) I PJJ-lémm Z [A n,j ()(JJH2 for every teT. 77,, (t) we know g a X2 has the decomposition a a E] + £2 and P lim t (t) a g (t) for every teT, 2v 2 v-avoo as was seen in the proof of Theorem 2.3.5. By Lemma 2.4.] “"‘l r lim 2. [A .(xnze §2(t) "'9‘” m(t) M for every tcT. 54. Corollary 2.#.3. Let [X(t), F(t); teT} and {Y(t), F(t); teT] be a.s. sample continuous second order martingales. If (TE; :1 2 l) is a sequence of partitions of T as defined in 2.2.], then P lim 2, A . (X) A . (Y) exists .19 m NJ ”1.1 m(t) for every teT and the process so defined, which we indicate by {z(t), F(t); teT] can be taken to have a.e. sample function of bounded variation and continuous on T. Moreover, if VQD) denotes the total variation of Z(-,u>) over T, E(V(u>)) < 00. Proof: We have ' "I Z. An,J(X)An,j(Y) . Z 114([An:‘j‘(v)+An,j 001244.»;MAM-(2012) fin“) fin“) . L n/uuamwanz- [Ame-W) . m(t) Let g - X+Y and E'- X-Y. Then E and E are a.s. sample continuous second order martingales, so £2 and E2 are a.s. sample continuous positive semi-martingales. We write fl fl 2 -.. 2 2" Amm Amwx) - L unmade-J] - L I/lllAn’jGH m(t) m(t) 7Tn(t) and hence, by Theorem 2.4.2, 55- ’ "'1 2 -2 P [1300 Z An’jwmmjm . I/Mlé 12(t) - [a 12m) . m(t) Let Z(t) - l/h([§2]2(t) - [E2]2(t)). Then the Z-process has the stated preperties since both [52]2 and ['52]2 have these prOperties. He now discuss some other results which will be used in Chapter III. Suppose (X(t), F(t); teT} satisfies the conditions of Theorem 2.3.5. Let [X]l - XI and [X]2 - X2. Then XI and X2 are a.s. sample continuous, X2(t) a P lim 21‘ cn’j(x), and if V0») is "* °° rrnm the variation function of X2(°,w) over T, E(V(w)) < to. Let [Y(t), F(t); t T} be an a.s. sample continuous process. Then for a.e. a» the Riemann - Stieltjes integral 1 ~n * R Y(t)dX (t) - li Y(t .)£s .(X ) * exists, where t . < t < t nu "' “,1 - "’1‘”. we will show I --. * afvumxzu) - 9 lim Z Y(t .)c J.(x) fin O n-D'oo It will be sufficient to show 1 --. RfY(t)dx2(t) - Pngmm Z Y(tn’j)cn,j(X) 0 ”n q I Since P lim 2:. Y(t .)C .(X) a P lim n—> 00 W n,_] ""1 n-> 00 n ' n * Y“n.1’°n.1 "0 AW; 56. ‘I' where t . < t < t . n.9J - "Ll — ”’1'” To see this, let ~Ar r. - [ sup |Y(t)- HS” 2 Hr] ’ |t-slsl/r' r,r' - l,2,.... Because of the a.s. sample continuity of the Y-process, lim P(~A .) - O for each fixed r. r'->co r,r Assume n is such that "11:1“ < l/r'. Then P(IIZ hug,» v1cmj|> e1) 77?. -fi * - tuxA IL (mm)- utmncwun > e1) r,r' fl; .., «u- + P(lx~A,.,...'2. (mm)- vun’jncmun > an n 5 p(~Ar’r,) + Us] '2. [Y(t:’j)- Y(tn’j)]Cn’j(X)ldP Ar,r’ ”n 5P(~A ,) + We .. l at _ or 7271”] mtnu') Y(tn’j)llcn’j(X)|dPl A'r,r’ §P(~Ar’r,) + 1/6 Mr 5% ”mm” . n If n > 0 is given, choose r such that l/er (Kx) < q/2, then choose r' such that P(~Ar I,.) < n/z. Then for all h with “17;“ < l/r', ’ .‘1 * Hug: Mum.)- Yuma] “n,j"‘" > e1) < n- n 57- He first assume the x -process and the Y-process are uniformly 2 bounded. Then P(IIZ vumuamuz) - cn’j1|>e1) IA I/ez an L mmna M-(xz) cj(x2)1|2> 7Tn .. we! 5% |Y(tn,J)IZIAn,J-(X2) - cn,j(x2)|2) TTn M: i S e7 5‘24 [|An,j("2”2‘ 'cn.J(x2HZD Trn M; .. S :2 E(ngu lAn:.i(x2)|Za IAHJO‘ZH) .. l'l 77’" 2 NY '< -—- ZS . - e2 “05'"?ng In.50‘2)”(‘”)) Now max IA (X2)|V(w) is dominated by 2M V(w) where "X =- N ”’1 "2 2 sup |X2(t,w)| and hence t,w T—im MHZ m .)A[nj(X2) - an.(x)]| >e1) 0")(0 r ) , n E( T1? max (X)IV(‘”)) ‘0 h->ao ogjgNn IA "’J 2 since max [An j(X2)| —> 0 a.s. as n —> to because of the 3 58. a.s. uniform sample continuity. If we now drop the condition of uniform boundedness on the Xz-process, we can define a sequence of uniformly bounded processes {Xv(t), F(t); teT) v a 0,l,... as in Theorem 2.l.3 such that P([Xv(t) :- X(t); teT]) —> I as v —>oo. For each v - 0,l,..., the Xv-process will be a quasi-martingale with [Xv]1 - xiv, i - l,2, being the xi-process, i - 1,2, stepped at rye»). The Xv-processes have all the preperties stated in Theorem 2.3.5. Still assuming the Y-process is uniformly bounded and a.s. sample continuous, we.have just shown that for each v - 0,1,... I -"—I R.[ Y(t)dX2v(t) - Pn-l-imoo% Y(tn’j)Cn’j(Xv)- n For a.e. u), there exists a v(u>) such that for all 1/ Z v(w), X2(t,a0 - X2v(t,a0 for every teT and hence I l lim fY(t)dX2v(t) = fY(t)dX2(t) a.s. ‘w€>oo () 0 If we now show that 21’v(tn,j)cn’j(xv) converges to 2:.Y(tn’J)C .(X) "3.1 77h 77.. uniformly 32.2) it will follow immediately that l .1 fY(t)dx2(t) .. 9 lim Z Y(tn’j)Cn’j(X) Ir€>oo ° 77?. Consider P([IZvumJncn’flxp - swam > e1) Trn ‘,n ...mo; “1 . t . 7.1... W. 5‘9- 5 P([-rvko) 'oo and immediately after the proof of Theorem 2.3.5 we showed the second term goes to zero uniformly in n as v —> oo . we have now shown I -.-. R{v(t)dx2(t) - nigger)? Y(tn’j)Cn’j(X) n when Y is uniformly bounded and X satisfies the conditions of Theorem 2.3.5. Suppose now Y is a.s. sample continuous but not necessarily uniformly bounded. we can define the sequence of uniformly bounded a.s. sample continuous processes [Yv(t), F(t); teT] v a 0,l,..., as in Theorem 2.l.3, such that P([Yv(t) a Y(t); tsT]) -—>' I as v ——>' 00. For each v a 0,1,... we have just shown 3‘1 l Pn-l-émoo 2" Yv(tn,j)cn‘,j(x) '3 Rva(t)dX2(t) n 0 when the X-process satisfies the conditions of Theorem 2.3.5. But for a.e. 4:, there exists v(w) such that for v 2 v(w), Y(t,o.>) a Yv(t,m) for all teT, and hence 60. l l lim va(t)dX2(t) fY(t)dx2(t) a.s. v-D'oo () 0 Then by Lemma 2.4.l, l P lim 2:. Y(tn’j)cn’j(x) a R\/fiv(t)dx2(t). fin né>oo 0 we have now proved the following theorem. Theorem 2.4.h: If the process (X(t), F(t); teT) satisfies the conditions of Theorem 2.3.5, and if {Y(t), F(t); teT] is any a.s. sample continuous process, then I -e * R[Y(t)dxz(t) . pnglgo szfltn’jfln’jfl) n *- where t u< t .'< t . . n.J- n.J -' n.J+| The next theorem follows almost immediately from Theorems 2.4.2, 2.#.h and Lemma 2.h.l. Theorem 2.4.5: Let (X(t), F(t); teT) be a second order a.s. sample continuous martingale. Let g a x2 and let [t]2 . g2. Then if [Y(t), F(t); teT) is any a.s. sample continuous process, P lim Zutljnamxmz a P lim ZY(t:’j)Cn’j(X2) 77} 'Tn lr€>oo n-D’oo .l a R~/‘ Y(t)d§2(t). o bl. Proof: 1 we know 9 lim 2m: .)c .(xz) - RfY(t)d§ (t) lr€>oo M’ n’J 2 by Theorem 2.#.h. Assume first that Y is uniformly bounded and X is uniformly bounded and a.s. sample equi-continuous. If we look at the first part of the proof of Theorem 2.h.2, we see immediately that :3me m {ll 12le -P 11m 002: Ht", ~ ”up X2) con.“ 1 RfY(t)d§2(t). 0 Now assume X is an a.s. sample continuous second order martingale and Y is any a.s. sample continuous process. we can find two sequences of processes (Xv(t), F(t); tsT} and [Yv(t), F(t); tET]; v - 0,l,2,... such that for every v - 0,l,... xv is uniformly bounded a.s. sample equi-continuous, Yv is uniformly bounded and P([Yv(t) a Y(t) and Xv(t) a X(t); tsT]) —> I as v ->'oo. Since X satisfies the condition of Theorem 2.3.5, so does Xv for each v a 0,l, we let [Xv l a] v H(t)d§2 (t) e 9 lim Z." (tn’j)[An’J.(Xv)]2 O n>oo1T ]2 a x2v for each v a 0,l,.... Then ”n for each v - 0,l,..., as we have just proved. For a.e. «5 there exists v(w) such that if v Z v(a>), Yv(t,w) - Y(t,w) and §2V(t,w) :- §2(t,w) for every teT. And hence 62. l vgmm R! Yv(t)d§2v(t) - RfY(t)d§2(t) a.s. Then, by Lemma 2.h.l, .., l P lim 2 Y(tn,j)[An’j(X)]2 - Rf Y(t)d§2(t) - 0 n né>oo Chapter III: Stochastic Integrals Section I: General discussion: In this chapter we will define a stochastic integral for quasi- martingales. The approach we use is that of limits in probability of Riemann - Stieltjes sums. Ito (5) and Doob (1, Chapter Ix) have defined stochastic integrals with reSpect to a particular type of martingale process. Doob assumes the martingale (X(t), F(t); teT] has the preperty that there exists a monotone non-decreasing function 6(t) such that if s‘)" step functions, (I, Doob, p. 1-526). If we let t Z(t) .. of Y(s)dX(s) 0 then the process [Z(t), F(t); téT] is always a martingale with t E(Z(t)) E o in t. Further, if 20(t) - of Yo(s)dx(s), then 0 t e)~ f E(Y(s)Yo(s))dG(s)- 0 These properties of the integral are very nice in applications (See,for example, section 3, Chapter IV of Doob). Unfortunately, the integral does not have some of the more common properties that one associates with the ordinary Riemann- Stieltjes integral. For example, with the Doob integral we have no integration by parts theorem. Furthermore, one of the major defects is the non-existence of a reasonable transform property. This is indicated quite easily by an example in Doob. If the martingale process (X(t), F(t); t T} is such that D jP[X(t) - X(0)]dX(t) exists, then it has the value l/2[X(t) - X(O)]2 - l. i. m. ”22mm (.X)]2 n+>oo TTn where l.i.m. indicates limit in quadratic mean. It is readily apparent that one cannot hope to obtain a theory of stochastic integrals which parallels Riemann - Stieltjes integra- tion if we use this definition of a stochastic integral. In the next two sections we will define a stochastic integral and give some 65. of its properties. Although the exposition is far from complete, it is haped it will illuminate the feasibility of obtaining a Riemann - Stieltjes type stochastic integral. Section 2: Definition of the integral. In what follows we will again be assuming T - [0,l]. Let (X(t), F(t); teT) and (Y(t), F(t); teT] be quasi-martingales with P([X(t) - X,(t) + x2(t): teTJ) a I P([Y(t) - Y,(t) + Y2(t): tell) - I where a's usual, [x1i - x and mi - v , 1 - 1,2. 1 i we assume X1 and Y1, i - l,2, are a.s. sample continuous. Let [TTB, n 21l} be a sequence of partitions of T as defined in 2.2.l. we will show that .2.] 2; l 2 . + t . X t . -X t (3 ) Pug”... /[Y(tn’J+,)Y(n,J)ll("u“) (Mn 17.. 1 exists and we define ~./\Y(t)dx(t) to be this limit. 0 we will write Ami") . mum“) - Yum.” and End") l/2[Y(tn’j+l) + Y(tn,j)] we will use freely the notation introduced in Section 2 of Chapter II. we can write the sums in 3.2.1 in the form 2. 75M m Ami 0‘) TTn First _i 2. EMU) Amjm - Z Emlvmmwxil + En,j(Y)An rTl'I 7T“ 7T“ .J “‘2’ ' Since Y has a.e. sample function continuous and X2 has a.e. sample function of bounded variation on T, the limit of the second sum exists a.s. and is the ordinary Riemann - Stieltjes integral of Y(-wn) w.r.t. X2(-,w). we indicate this as follows 1 ’fi (3.2.2) Rf Y(t)dX2(t) . P lim Zzn,j(Y)An,j(x2) . n-> oo 0 TTh Consider now 2 Emmemw - bump/smug + Z l/2 An’j(Y)An’j(X,) . 1T" "n m The second sum on the right can be further reduced to the following 21/2 Andnlmn’jml) + Z 1/2 An,j(Y2)An’j(Xl) . TTn 7Tn Again, since X1 is a.s. sample continuous and Y2 has a.e. sample function of bounded variation on T, the second sum goes a.s. to zero. we have now reduced the problem to showing the existence of the limits '"fi P H. Ll 2A . Y A . X P lim Z'Y t . A . X "_>moo / n,J( l) n,J( l), n->oo (n’J) n,J( l) 1T" n It was proved in Corollary 2.4.3 that if Y] and XI are.a.s. sample 67. continuous second order martingales, then " 2 2 21%.“; Z l/2 en’jnlmmj (x1) . l/amla. "th' [(vl- X9120». fin the limit being a.s. sample continuous and having a.e. sample function of bounded variation. If (Y'v(t), F(t); teT], [le(t), F(t); teT], v - 0,1,... are as in Theorem 2.l.3, then by Theorem 2.l.l each xlv and Y are uniformly Iv bounded a.s. sample continuous martingales and P([Ylv(t) + Yl(t) or le(t) + Xl(t) for some teT])«-€> 0 as v -—> 00. Hence by Lema 2.ll.l Plim 212A .Y A .X exists. "_m / Mm mu) 77’. _. . 2 Further, if §2v :- [alv-l- lez, §2v -_ KYW- Xwflz, then “ ""l (3.2» Pngmml Il2An,,i(Yl)An,,i(x|)nPvlémm1/8(§2\I(|)”gnu”. 'Tn we define ‘_l l . (3.2.5) f dXdYa P lim 2) 1/2 An,j(Yl) Arum.) . 0 n€>canT n If we now show P lim Z Y(t .)A .(X ) exists, we will have n:J n)J l n-¢’OO 77.. proved the existence of the limit in (3.2.l). To prove this we first prove the following lemma. Lemma 3.2.6. If {X(t), F(t); teT} is a second order martingale and 68. [Y(t), F(t); tsT] is a uniformly bounded, a.s. sample continuous process then \_ fl lim Z. Y(t )A (X) exists in quadratic mean. n->oo ”1] 0).] n Proof: Let m > n. Then IT" c: TTm and we can write zn - ZYe >...>0andlim s so. Also let [a ., r'_>_01bea O l r r r->oo . sequence of positive real numbers with 60 > e] >... > o and ‘1'" 6 I‘ 00 r-L>cn Define “’Ar r.(t) -[ SUP |Y(5) - Y(s')| fer],r,r' =- 0,l,2,.... ’ ls-s'lié,- s,s' st 69. Because of the a.s. uniform sample continuity P(Ar r.(t)) —> 0 as r' -->oo for 3 every fixed r. Now for fixed r and r', if tI > to then r'(tl)C ~Ar,r'(t0) and hence r'(tl) :3 Ar’rl(to) when tI > t0 . Z, Z. E(IY(tnmj’k)- Y(tn’j)| IAnmj’k(x)| ) Trn 7TH .. L f |Y(tnmj,k)' Y(tn’j) |2|Anmj’k(x)|2dl> 77’. mj Ar,r'(tnmj,k) -_1 4. EN .(t "2... fr IY(tnmj,k)" Y(tn,_j)lzlAnmj,k(x)|2dp ° nmmj nmj,k) Let MY - sup lY(t, w)|' Then 2 2 "Rheum" Y(tn,j)| lAnmj,l<(")I d" Ar, r' (tnmj,k < AMY f IA Am”. k(x)|2cll> l’|"(tnmj,k) - 4N2 |x(t )| 2d? - L. 2 pm: )| 2d? Y nmj, k-l-l NY nmj, k r,r'(tnmj:k) rr, r'(tnmj,k) 70. 2 2 2 ' 2 - l"ll! f lx(tnmj,k+l)| d" ' ““y _[ NtmLkll dP - Ar,r (tnmj,k+l) Ar,r'(tnmj,k) A Z. 2.. f |Y(tnmj,k)’ Y(tn,j)|2lAnmj,k(x)|2dP Trn 7I’hmj Ar,r'(tnmj,k) 5 1m: j |X(l)|2dP Ar,r'(') Let c > 0 be given, choose r such that 6'2, E(|X(l) - X(OHZ) < e/2. Since P(Ar '-.(l)) —> O as r' —> 00 for every fixed r, we can now , . choose r' such that 1m: f ( |X(l)|2dP < s/2. Now choose n(e) A I) r,r' such that “7121(8)“ < er“ Then if m > n 2 n(e) 2 2 2 2 2 l5(|z"I - an ) _<_er E(|X(l) - x(0)| ) 4» MY] |x(l)| dP Ar,r'(l) S e/Z + e/Z :- e. The lelmla is now proved. If [X(t), F(t); teT] is any a.s. sample continuous martingale process and if (Y(t), F(t): teT] is any a.s. sample continuous process then P lim 2 Y(t .)A .(X) exists . ".900 nu n.J 77.. For, if {Xv(t), F(t); teT) and {Yv(t), F(t); teT] v 20, are as defined in Theorem 2.l.3, then Xv is a uniformly bounded a.s. sample continuous martingale and Yv is also uniformly bounded and a.s. 7]. sample continuous. Then by Lema 3.3-5 "limm Z Yv(tn,j)An,j (Xv) exists in quadratic mean for 77.. every v 2 O, and hence by Lelmla 2A.] "1 9 l1... 2' Y(t .)e .(x) exists. "_>m nu nu 77.. we now define 1 (3.2.7) o] Y(t)dX (t) - 9 lim Z m .)A .(x) . l n->m n,J n,J l 0 77" n we have now established the existence of the limit in 3.2.] and we define l l l .l . (3.2.8) jvmdxm - DfY(t)dx‘(t) + RfY(t)dX2(t) +de(t)dX(t) 0 0 0 0 n-> oo - 9 lim 2 Em") emu) . Tl'n Clearly, if [a,B] is any closed sub interval of T, we can define .5 .-—, (3-2-9) j Y(t)dX(t) = P gm 2. En j(Y) (kn-100, m 2 I u n 77.05:" for the limit will exist when (TI-GB n’ n _>_ I) is a sequence of ’ partitions of [u,B] with ”Trafimn -n—> 0 and 77:15,]: ”35,2: . If UT", n 2 I) is a sequence of partitions of T as defined in 2.2.1 and if 7Tn(t) -T)’nn [0,t], then t ""1 (3.2.10) Y(t)dX(t) =- P lim Z an j(Y)An J.(x) o "'>°° m(t) ’ ’ 72. Section 3: Some properties of the integral. Let (X(t), F(t); teT] and [Y(t), F(t), teT] be quasi-martingales with I P([X(t) - x,(tl + let); teTJ) - l P([Y(t) - v,(t) + Y2(tl; teTJ) - l where [X]1 - Xi and [Y]1 - Y1, i . l,2, are a.s. sample continuous. t Let Z(t) - jY(s)dX(s) - 9 lim Z Emu) emu) . 0 n-D’oo 773(t) Theorem 3.3.l. The process {Z(t), F(t); teT}, as just defined, can be taken to be a.s. sample continuous. Proof: t t t Z(t) -fY(s)dX(s) .- DfY(s)Xm(s) + RjY(s)dX2(s) 0 O 0 5t +de(s)dX(s) . 0 C First, the integral R‘/\Y(s)dX2(s) as a function of its upper limit 0‘ defines for a.e. a» a real valued continuous function on T. t a The integralde(s)dX(s) =- P lim Z l/2 A .(Y ) A n,j l n, -> oo . (XI) 0 " m(t) J a P lim v-D’oo '/8(§2v(t)‘ §2v(t)) Where §2v and §2v are as defined in 3.2.4 is a.s. sample continuous by Theorem 2.2.10 since both §2v and 32v are a.s. sample continuous for every v a l,2,.... It thus remains to show the integral t --. DfY(s)dX (s) - P lim 2 Y(t .)e .(x1) 0 n->mTrn (t) as a function of its upper limit defines for a.e. an a real valued continuous function on T. we first assume the XI and Y processes are uniformly bounded and a.s. sample equi-continuous. By Theorem 2.2.l0 it is sufficient to show that given e,q >'0 there exists n(e,q) and B(s,n) such that for all n 2 n(e,n) (I) P(ll 52.255“ ”Z“ Y(an’jlxll-Z‘ Y(tn’j)An’j(X,)l><-:]) ’" m(t) . TT (5) n < n First we show that if e,q > 0 are given, there exists an n(e,n) such that if n 2 n(e,n), then for all m _>_ n, P v t (x) ' l) “05;"... OSmfoSknmj I11: ( MU 1M nmj k ' |>e < 11 Now P( Y(t )A . (x) e]) [0 232"" Osngknmj '12-?) nmjd. nmJ.i l I) s P([OSmjagNn 053W lgolmnmj’i) "Y‘tn. J.)1en,,u.,kI > e121) + ”[033. ngiknmj “(tnuml‘tnmhml’ ' xl“n.J”'>‘/2]) we have P([ max max |Y(tn j)(Xl (tnm jk+l)' Xl (tn j)” >e/2]) 0'e 2 — ([0 '("’J)| IHYJ) - - n '- - nmu where HY - sup|Y(t,u9| . t,ul Because of the a.s. sample aqui-continuity of the Xl-process we choose a 5 >’0 such that P([ sup le(t) - Xl(s)| >te/2MY1) - 0 . l lt-sl $51 If "1(6) is such that for n 2 n](e), IlTrnll< 5], then P([ max max IXl(t )-X(t )>€/2 ])-O. ogj-gun09ngJ ' ' "’3' HY . nmu,k+ Observe that the partial sums k Zlfltnmj’k) - v’ N n _i k s, P([ max I LIV/(tnmj’kl-Y(tn’j)]Anmj’k(X,)l> e1) j-O 05 k-<"nmj 1-0 N k nmu 5 l/e2 : z E(|Y(tnmj,k)- Y(tn’j)|2|Anmj,k(Xl)|2) . j-O k-O Now choose 52(e,q) such that 2 SUP |Y(t)" Y($)l2 _<- 6 2 < T] a.s. It-slsaz(e.n) E(lxl(l)-X,(0)I ) Then choose n2(e,n) such that ||1Tn|| < 52(e,n) for every n 2 n2(e,q). 75- Then Nn knmlj [62 Z. 2.: E”Y(tnmjflt)" Y(tn J” ZAI nmj, kO‘IHZ) < 1] . j-O k-O Hence if n 2;max {nl(e,n), n2(e,q)], we have the desired result for all man. we can now show (I) is true. Assume m >ln and 5.( min It '+l- t .|. Then OSJSNn n)J ”Li (2) ml sup I23»: (mt .110" 2: Y(tm1)Am,(X)l> 3e] ”5'55 mylt) mum _<_P([ max max IZY(tnmj,i)Anmj,i(xl)l>61) 05:5". 05 kit... 1.. If n(e,q) a max {nl(e,n), n2(e,n)] is as chosen above, then we can fix n 2 n(e,n) and let 5(e,n) _<_min ltn,j+l- tn,j|' Then (2) will be less than n for all m 2 n. Hence .., t Pngmm 2 Y(thngo‘l) .- DfY(s)Xm(s) m(t) o is a.s. sample continuous when Y and XI are uniformly bounded and a.s. sample equi-continuous. The desired result now follows by stopping Y and XI according to the stOpping time defined in 2.3.3 and then applying Theorem 2.2.l0 and Lemma 2.4.l. with what has already been shown, we can easily obtain conditions under which the process Z(t) -‘/fi Y(s)dX(s) is a quasi-martingale. ° 1 76. Theorem 3.3.2. Let {X(t), F(t); teT) and {Y(t), F(t); teT] be quasi-martingales. Let [X]1 - X1 and [Y]i - Y1, i - l,2, be a.s. sample continuous. Further assume Y is uniformly bounded and Y1 and XI are second order martingales. If t Z(t) -~/fiY(s)dX(s), for every tsT, 0 then the process (Z(t), F(t); teT] is a quasi martingale with t ' t l: min) - DfY(s)Xm(s) [212m - RfY(s)dX2(s) +de(s)dX(s) 0 ' O O and [Z]1 - 21’ i . 1,2, are a.s. sample continuous. Proof: Since Y is uniformly bounded and X] is a second order martingale, by Lemma 3.2.6 t i... o Y(s)Xm(s)- lim 2 Y(t .)e .(xl) {f' n-D'oo 773(t) n J where the limit is in quadratic mean. So ZI being the limit in quadratic mean of a sequence of martingales is again a martingale. we have further shown, in Theorem 3.3.] that 21 is a.s. sample contin- t uous. Now R /qY(s)dX2(s) defines a continuous real valued function 0 of bounded variation on T for a.e.lm when considered as a function of its upper limit. Also t ‘fi de(s)dX(s) =- P lim 2. 1/2 An,j(Yl) An,j(x‘) 0 "9 °° 7Tn(t) a l/8([(X‘+ Y])2]2(t) - [(xl- Y,)2]2(t)) 77. is a.s. sample continuous and of bounded variation on T since XI and Yl are second order martingales. Therefore 22 has a.e. sample function continuous and of bounded variation on T. The theorem is now established. we now investigate some prOperties of the integral which parallel the Riemann - Stieltjes integral. Theorem 3.3.3: Let (X(t), F(t); teT] and (Y(t), F(t); teT] be quasi-martingales with [X]i - X1 and [Y]j - Y i - l,2, a.s. sample continuous. Then I l fX(t)dY(t) +fv(t)dx(t) .- X(l)Y(l) - x(o)v(0) a.s. 0 0 1) Proof: Observe A.XY-E.XA.Y+A.YA.X sothat n”(l n”(l n,J() n,J() null. 24 Anni (XY) - 245")..l (X) An1.j(Y) + 24 any] (Y) Anni (X) Trn 77h 77;. Hence, taking probability limits on both sides we have the desired result. One thing that one would expect of an integral is the following: (X(t), F(t); teT] is a quasi martingale and if the function f is such that t f f ' (X($))dX(S) exists 0 t then d/Nfl(x(s))dx(s) a f(X(t)) - f(X(0)) . 0 78. For the Doob integral this is not the case, as is illustrated by an example from Doob (l, p. ##3). If [X(t), F(t); teT} is such that OJ/‘[X(t) - X(O)]dX(t) exists, then ojiuu MMNMQ-IRUU)XWH - l/2 l1 X mngJAnuln n where the limit is in quadratic mean. Let (X(t), F(t); teT] be a quasi martingale with [X]1 - X1, 1 - l,2, a.s. sample continuous. Let f(t) - t2. Then l X(l)2- x(o)2 - f2X(t)dX(t) 0 or I “unl-Huml-jVWHQMMu. 0 """1 ‘ — For, f2x(t)dx(t) - 2 Pnlimoo 2 1/2 AM. (x) emu) ° ' TTn - P lim 2‘ An’j(X2) - x(l)2 - x(o)2 n-D’oo n This property of the integral can be generalized to the following extent - Theorem 3.3.h. Let (X(t), F(t); teT] be a quasi-martingale with [X]i - X1, 1 - l,2, a.s. sample continuous. If f is a‘real valued function of a real variable and has a continuous second derivative, then nun>~umm>1flfwuammn- 79- Proof: we want to show f(X(l)) - f(X(O)) - P u... 2 Kmuwxnen’ju) n-> oofl.n - 9.1:“... 27 “(X(tn’jlmmjlx) + Pulimm2fl/2Amulxnemm n n T . 7‘ .. * 2 ' Peg”... 2n" f (“humus“) " 9.310;; "2 f ““n.1’““n.3"‘” n n * a where t . < t . < t . and where t . de nds on a). “U - NJ "' "2.1"" "U [)6 we can write «whamm-Ziflmn Tl'n ‘-1 2 [f'(X(tn, 1))An, j (x) + 1/2 f"(X(t:’:))lAn,j (Xllzl lTn -)l- il- -X- where again t . < t . < t . , and where t . depends on 03. NJ "' n1J " n2J+l nyJ Hence, |f(X(l)) - f(X(0)) - EM f'(x)An’J.(X)| 3M - I/ZIZ [f"(X(t:,J.)) - f"(X(t:’:))llAn’J-(X)]2| "n _<_ l/2 2; qufidn - f"(x(t:.,j))||en,j 0 a.s. as n -—>oo. If Z [An j(X)]2 converges in probability, then ) lTn End (f(x))An,j (X) will converge in probability to f(X(l))-f(X(O)). 3W] Now 2 [AM 0012 2 [AM-(2(1))2 2%”. (xllem (x2) 77.. 77h 77.. +2 [Am (xznz- 77.. Because of the a.s. sample continuity of the processes XI and X2 and the a.s. sample bounded variation of the Xz-process, the second and third sums go to zero a.s. as n ->'oo. we have previously shown the probability limit of the first sum exists. The theorem is then proved. A natural question at this point is what additional assumptions on the function f or the quasi-martingale X will insure the process [f(X(t)), F(t); teT) is a quasi-martingale. Assuming f and X satisfy the conditions of Theorem 3.3.#, we have t mm» =- f(X(0)) +ff'(x 007Tn(t)n’J n’J ' _[ Consider then 2 En’jfi'lxnamjml- Z P(xltmllamlxp 77"“). 77h“) ~ + l 24A . f' .A X . Z I ml ()0) M( I) rrnlt) Assuming f' is bounded and continuous, the first sum converges in quadratic mean and therefore defines a martingale process. Last of all, consider 2 WA .(f'(X))A (an-Z l/2f"(x(t*.))[A (H12. naJ “2J1 ")J "Ll I 1113:) m(t) when X' is an a.s. sample continuous second order martingale, we showed in Theorem 2.“.5 that -., t P 11m 2 1/2 mm: .))[A (Xllz-ff"(X(s))d§(s) :J "1.1 l where g - [Xf]2, provided f" is continuous. Then for each teT, t t f(X(t)) - .‘(x(0)) + fo'(X(s))Xm(s) + Rff'(X(s))dX2(s) 0 O I t +ff"(X(s))d§(S) 0 82. If we let t '[f(x)]l(t) - O‘/fif'(X(s))Xm(s) for every teT, and 0 t t [f(X)]2(t) - mm» + Rff'lxlsndles) +ff"(X(s))d§(s) for 0 0 every teT, then the process [f(X(t)), F(t); teT) is a quasi martingale if the above conditions are satisfied and E(|f(X(t))|)‘< 00 for every teT. we summarize these results into the statement of the next theorem. Theorem 3.3.5. If f is a real valued function of a real variable with f' bounded and f" continuous, and if (X(t), F(t); teT) is a quasi- martingale with [X]1 - X1, 1 - l,2, a.s. sample continuous and XI second order, then [f(X(t)), F(t); teT] is a quasi-martingale if E(|f(X(t))|)< 00 for every teT. Further t [f(X)]l(t) - 02/‘f(X(s))dX'(s) for every teT 0 t t [f(X)]2(t) .- f(X(0)) + Rff'(X(s))dX2(s) + Rff"(xls))d§(s) 0 " 0 for every teT, where g - [X12]2 . Assume now (X(t), F(t): teT] and (Y(t), F(t): teT) are quasi- martingales with [X]1 - X and [Y]1 . Y i - l,2, a.s. sample 1 1’ continuous. Assume further Y is uniformly bounded and XI and YI are second order martingales. 1: If Z(t) - “/‘Y(s)dX(s) for every teT, then by Theorem 3.3.2, 0 we know the process (Z(t), F(t); teT] is again a quasi-martingale 83. with the following decomposition. t [Z]l(t) - Zl(t) .- DfY(s)Xm(s) 0 t [212m - 22(t) - RfY(s)dX2(s) + 1/8(§(t) - F(t)) 0 where t - W1" vllzlz. E- [(x,- Yllzlz. Let f be a real valued function of a real variable with f' bounded and f" bounded and continuous. Then since ZI is a second order martingale and 21, i - l,2, are a.s. sample continuous, by Theorem 3.3.4 , t f(Z(t)) - f(Z(O)) +~/' f'(Z(s))dZ(s) for every teT. 0 we wish to show that l l f(Z(U) - f(Z(O)) -ff'(z(s))dz m n Consider 2 En,j(f'(z)Y)An,j(x) . 2 En,j(f'(z)Y)An,j(x2) "3 I7}. .“1 .2 f'(Z(tn’j))Y(tn,J.)An’j(XI) + ZI/ZAn’j(f'(Z)Y)An’J.(XI) . 77.. Tl’n The third term can be rewritten as follows: 84. 2: l/2 An’j(f'(Z)Y)An,j(xl) . 2' l/2Y(tn’j+l)An,j(f'(Z))An’j(Xl) Tl’n 7T}, . + m f'(2(tn’j))An’J. (mm (x,) 1/2 Amman’j (f ' (zllam. (x,> 3M SW1 4- Z, 1’2Y(tn,j)An,j(f.(z))An,j (XI) Tl'n + Zl/Z f'(z(tn TE. we will show that .1”An.j (”Am 0‘1) ' I) Pngmmz'f.(2(tn2j))Y(tn2J)An2j(x1) lTn .- l f' z . A Phi"... Z ( (tn’Jl) TTn .(ZI) n2J 2) P lim [2 En .(f'(z)ll)An .(xz) +Z’l/2 f'(Z(tn j))z.\.n j(WAn 10‘1” n->co 2J 2J 2 2 2 7T“ Tr" l - 213.00% En’jlf'lzlmn’juz) - R~off'(2(t))d22(t) n 3) Pnlimm Z l/Z An,j (f' (2))Y(tn,j)An,j (XI) 77}1 a P lim Z, ”2 An J.(f'(Z))An j(zl) n->m ’ ’ 77h 85. and finally '4) Pngmm 1211‘ l/2 And (f-(z) )An,j (”A021 (XI) - O . n To show I) is true, consider sq2f'lzltn,jllv(tn,j)amj(xp -Lf'(2(tn’j))An,j(Z,)|2) Tln 1T" E(l 2 f'(Z(tn’j))[Y(tn,J.)An,J. (x,)~ AM. (2,)1I2) 77h E(Z [f'(zumjlllzlvltmlem (x,>- AM. (2912) T7; 5 hi. E(Z[Y(tn’j)An’j(Xl)-An’julnz) Tl’n where Hf. - SUP lf(Z(t.w))| - t,u> u... E(IZIU) - 2 Y(tn’jmn’jmllz) TTn . E(IZ An’j(zl) ->_.Y(tn,j)An,j(xl)I2) TE. 7Tn ' . m2 [Am-(Zn) - Yawn“ (x,)1|2) m. 177. and l) is now proved. 86. To prove 2) we first observe that Pn-l-imm Z 1/2 1”(Z(tn,j))An,j(Y)An,,.i(xl) TTn ' Pnlimaogfm f'(z("n.l”“n.1"I’An.1"‘l’ ' n Then we can write I l .t Rff'(Z(t))de(t) - Rff'(2(t))d(kfv(s)dx2(s) + 1/8(:(t)- F(t))l ‘ o o 0 l l . aff'(z(t))v(t)dx2(t) + 1/8 Rff '(Z(t))d(§(t)- f(t)) - 0 0 ' — out we Rff'(Z(t))d(t(t)- §(t)) 0 . Pngnmz l/2 f'(Z(tn,j))An’j(Yl)An,j(X‘) We and l ' ‘fi __ «[f (Z(t))Y(t)dX2(t) - P“l;m°°rr A... jlflzlvlem 1(le- n Hence 2) is proved. Recall f" is bounded and Y is uniformly bounded. Now to prove 3) consider, P li 212‘ 2.A. . f' 2 t .A m / M( (”H“) nnd>oo Tl}. n,j(xl) :--l * . P430021” f (Z(tn jnem. (Z)Y(tn’j)An’j(Xl) n i H. II * a Pngmm Z l/2 f (Z(tnduand(ZI)Y(‘n,3)An,J-(XI) . m. 87. Now El] Z In f"(Z(t:’J.))An’J. (gunmen), (XI) T"n -24 l/2 f"(Z(t:’j))[ z'(tn’j)12|2) T7}. . £(| 2 1/2 Hemline“agenda.)- Y(tmjmn’jlxmlz) n; 5 all 2 [1/2 f"(2(t:,J.))AM(2,)121”2 7Tn IZEAMXZQ- vltn’jlen’j(x,>12l"2> 1r. 5 EIIZ(Z [1/2 mm; J.))An’ J.(z,>12l 77:. E"2(Z[An’j(21)- Y(tn’leMJXIHZL W71 5 1/24,... t"2<2 [AM-(2)12) 6’22 [AM (2,)- Y(tmmn’j ”n 77.. l/ZMf" £"2(|zl(l)- z](o)|2) El/2(Z[An’j(ll)- Y(tn,j)An,J.(xl)]2) 7Tn where Hf" I sup If"(Z(t))|- 12,0 In proving l) we showed the term on the right goes to zero as n -€>'oo. Hence 3) is proved. 88. To show h) we observe 9.1;“... 2: 1/2 AM. (F(t))an’j mam 1x.) 17.. .- Pnl-imm % l/ZA n,j (f ' (Z) )AnJ (Yl)An,j (XI) n - Phil; 1/22 P(zumwllamj(human "h - P lim 1/2 Zf'(z(tn’j))An’j(Yl)An’-j(X') MG) m. .l _ ,l - I/8jf'(z(t))d(§(t)- ml) - 1/8jf'lzltlld1tm- F(t)) - o. O 0 we can now prove the following theorem. Theorem 3.3.0. Let {X(t), F(t); tcT] and [Y(t), F(t); teT) be quasi-martingales with [X]i - X1 and [Y]1 - Y i - l,2, a.s. sample continuous. Let 1) f be a real valued function of a real variable with continuous second derivative. If t Z(t) a ‘jp Y(s)dX(s) for every teT, then 0 l f(Z(U) - f(Z(O)) -f f'(Z(t))Y(t)dX(t) . 0 Proof: Let v a l,2,.... Let Tvub) be the first t such that 89. sup |Z(s,w)| 2 v or sup. |X(s,w)|2 v or sup|Y(s,m)| 2 v. sgt sit SSt If no such t exists, let Tv(w) = I. Since X, Y and Z are all a.s. sample continuous P([Z(t) + Zv(t) or Y(t) + Yv(t) or X(t) + Xv(t) for some teT])«-é> o as v -D'OD. Clearly, 7v is a stopping time for each of the processes X, Y, and 2. Let Xv, Yv and ZV be the processes X, Y, and Z stapped at TV. Then for every v - l,2,..., Xv, Yv, and IV are uniformly bounded by v and are a.s. sample continuous. we first show, for every teT t "'1 zvm - fv (s)d)(’(s) - P lim 2.5.. '(Yan J.(x ) n->'oo o m(t)" t * Let Zv (t) -~/‘Yv(s)dxv(s). For each teT, we can find a subsequence 0 of partitions (nut); k 2 I] such that 2:(t)-lim Z " .(Y)A .(x) a.s. Ak2J V k2J V and Pfi - Z(t) 1- lim 2‘ Ak,j (X)Ak,j (X) a.s. '9‘” m(t) If t 5 TV (cu), then z:(t-li ".vA.x) ) k_>...m2f'(t)e 191‘.)an = lim 2 5k .(mk .(x) .- Z(t) - z (t) . k—>OO”|u<(t) 2J 2J V 90. If t > 'rv(w), then 2*(t)-lim Z" Mme (x) k-> mflL(t)A kJ2 k2J .11... Z denlek’jm k“) “Wm-(ml) Eli"; Knievel)” Ak, “1.99”“? - Z(TP‘“)? - zylt), _. * sinceklimm Ak,j(Tv(al))(Y)Ak,j($(w))(x) :- 0. Then Zv(t) :- Zv(t) a.s. for every t€T, so that Z(t-Pli A (x . ) n->mm%i(t)- "2J( ”AV "2] ) Actually, one can take * P([zv(t) - zvlt); teT. v - 1.2.---]) - I. for we get equality on an everywhere dense subset with probability one, and since 2: and Zv are a.s. sample continuous they must be equal for every teT with probability one. Since for each v - l,2,..., Zv is uniformly bounded and f, f' and f" are continuous, f(Zv), f'(Zv) and f"(Zv) are uniformly bounded. But then l «2,11» - f(Zv(0)) - ff'(zv(t))Yv(t)dxv(t)- 0 For a.e. «3, there exists 12(0)) such that Tv(u>) :- l for all v Z v(u>), and hence I f(Z(l)) - f(Z(O)) .. ff'(Z(t))Y(t)dX(t) a.s. 0 9|. The theorem is now proved. Further properties of the integral need to be investigated extensively. Some of the theorems can be generalized somewhat, but in an obvious way. Some of the more pertinent questions which have not been looked into to any great extent are the following: 1) 11). iii) Can the decomposition Theorem 2.3.5 be extended to processes (X(t), F(t): teT] having a.e. sample function right (or left) continuous? Here it is felt that condition 1) of Theorem 2.3.5 may have to be replaced by the condition of uniform integrability of the sequence of flowing times (who); v Z l), where Tv(u:) is the first t such that sup |X(s,m)| 2 v . And if no such 5*< t t exists 3(a)) - I. what functions f of a quasi-martingale (X(t), F(t); teT} will again be a quasi-martingale? In terms of boundedness and differentiability conditions on f it is felt that in general f(X) need not be a quasi-martingale if f does not have a second 'derivative. If one investigated f(Z), where f is the integral of the wierstrass function and Z is the Brownian motion process, one should get some indication of whether the second derivative of the function f is necessary. One could also look for other conditions on f, such as convexity. what processes are integrable with respect to a quasi-martingale? Theorem 3.3.h indicates that the integrand may not have to be a quasi-martingale. 92. iv) Doob (I, pp. 273-291) has given solutions of the diffusion equations on the real line. with the definition of a stochastic integral given in this thesis, can the diffusion equations be solved on a sufficiently differentiable manifold, possible a twice differentiable manifold? It seems entirely possible this is the case and should be possible with relative ease. [ll [2] [3] [4] [5] BIBLIOGRAPHY DOOB, J.L. Stochastic Processes. wiley, New York, l953. LoEVE, n. Probability Theory, 2nd edition, D. Van Nostrand, Princeton, l960. MEYER, P. A Decomposition Theorem for Supermartingales, Illinois J. Math. 6 (l962), pp l93-205. JOHNSON, c. and HELHS, L. L. Class O Supermartingales, ANS, Vol. 69, l, pp 59-62. ITO,'K. Stochastic Integral, Proc. Imp. Acad., Tokyo 20 (lsuh) PP 519-52u. 93-