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Some Aspects of the Theory of Stochastic Integrals I Vietnarn Journal of MathemeGics Volume 23, Number 1, 1995 SOME ASPECTS OF THE THEORY OF STOCHASTIC INTEGRALS I DANG HUNG THANG Abetract. The aim ol the pper ia to gioc a biel suroeyon Eome dhzctiow ol rccearchintlu tluory ol etahutic intt4rud,ionutithtlu emphuic on aomn.topica rc.Iddto our work, Many imprta nt stelwtic il&egdt, uhich arc cerrtml intlu mdern theory ol stu,hafiic atulysis ard cnrcial lor apflicdioru arc prc,certed Tlle rcndom operdarc bctwecnBarw,h sryes, a rwtumlfrcmcwo* of stu,hastic intqrds, arc also trcd,ed. 1. INTRODUCTION The purpose of the paper is to give a brief survey of some aspects of the theory of stochastic integration and to present some of our con- tributions in this context. We does not attempt to present all aspects as well as the comprehensivehistory of the developmentof the theory of stochastic integration but rather just some directions of researchwhich are of our interest and related to our work. Historically, the first stochastic integral in the probability theory is the integral of a square integrable function with respect to (w.r.t.) the Wiener proces. This integral was introduced by N. Wiener in 1923 [Sf] and called the Wiener stochastic integral. The Wiener stochastic integral has been generalized in many directions. Genarally speaking, the aim of these generalizations is to define stochastic integrals so that the class of integrators as well as the class of integrands must be as wide as possible and, at the.same time, the stochastic integral should enjoy many good properties. Several stochastic integrals ( e.g. the Ito integral, the Stratonovich integral) are central in the modern theory of stochastic analysis and crucial for many other applications. 1 Research is supported by the National Basis Research Program in Natural Sciences and was written during the author's stay at Department of Mathematics, University of Amsterdam (Netherland) in September 1993 Dang Hung Thang Sections 2, 3, 4 and 5 treat some directions of research in the theory of stochastic integrals and in the last section we introduce the notion of random operators in Banach spacesas a natural framework of stochastic integrals. 2. STOCHASTICINTEGRAL OF NON-RANDOMFUNCTIONS WITH RESPECT TO A RANDOM PROCESS Let {W(t), 0 S t < 1} be the Wiener process(or the Brownian motion) on the interval [0,1]. In many applications, there arises the need of considering the integral of the form I f lf(t)dw(t), (1.1) J 0 where /(t) is a function defined on [0, t]. Because all sample paths of the Wiener process have unbounded variation, the Stieltjes integral I f I t(t) dW (t,tr) can not be performed on every path. However,Wiener J o wa.ssuccessful in finding a reasonable mathematical definition of the integral (1.1). The construction of the Wiener integral is the following' First, if / is a simple function IU):f ';11',,',*,;' d:l I f then / f (t)dw is definedbY J 0 I f" tttldw :lc;{w(\+,) - w(t;)}' JI o d:l By the independence of the differences we have the identity l1 tf P f ^ El I f@dwl: Ilf(t)rdt. IJ I J 00 Theoty of sbochutic integtels 1 This proves that the mapping / --+ t O* is an isometry from the t[ linear space S of simple functions into Lz@). Next, becauseS is dense in Z2[0,1] this map adrnids an extensionto the whole space LzlC.,ll. f For eachf e t r[O,tl, t(t) dW is definedas the imageof / underthis J o map. I In other words, f dW canbedefined as the limit in the mean tI Ol square of the integral sunilr of the form i r tr,l(w (t ;+,) - w(t ;)) r:o when the gauge of the partition tends to zero. By a similar construction, H. Crame developed a general theory of stochastic integral w.r.t. a complex-valued processwith orthogonal increaments. Recall that by a processwith orthogonal increaments we mean aprocessX(t),r€R satisfyingElX(t)12< m and Elx(r2)-x(r')llx(tn) -x(rr)l -s, whenever (t t, t z), (rs, ta) are disjoint intervals. It is shown that I f VldX exists for complex-valued functions / J R satlsfying f^ dG\ ( oo, J l|(tlf R where G is a non-decreasingfunction associatedwith X by the formula Elx(t)- x(")l' : G(t)- c(") and the following isometry holds Dl.f n: f dcu). J taldxl" J lf@i2 RR Deag Ilung ?hutg This integral plays a pivotal role in the theory of stationary processes. Namely, if X(t) is a stationary processthen there exists a complex- valued processwith orthogonal increamentS(t) zuch that f X(t): l"""ds(u). { This forrnula is called the spectral representation of the stationary pro- cess X{t). As a continuous analogue of weight sum of i.i.d random variables the stochastic integral w.r.t. a Levy process wa.s of the form fa;f; firstly studied by Levy and developedby Urbanik and Woczynski [34]. Recall that a process X(t) is said to be a Levy process if it has inde pendent increa,mentsand the distribution of X(4_- X(s) depends only - - the on t s. If t(t) is a simple function, /(r) fc;11t;,r;+,) then i:O I stochasticintegral I p1O*1t) is definedby J o I t n-1 I f (t)dx(t): - x(tr)]. J I ",[x(r,*,) 0 d:O A function / is said to be X-integrable if there exists a sequence(/,") of simple functions such that /"(t) convergesto /(t) almost everywhere a^ndthe sequence {-d' [ ndX] .onverges in probability. In this case we put l1 I -,., ._-,^,- f . - I t{t) dx(t\ P - t{" / I"(t) dx(t). J NJ oo It was shown that the'limit, if it exists, is independent of the choice of a particular appro:cimating sequence (/"). The following theorem gives a firll analytic description of X.integrable functions. Theorem (Urbanik,Woyczynski I34l). Let X(t) be a symmetric Leay - processond thc chardcteristie lunction 6(u) of the increoment X(t) X(s) 6e giuen by Theoty of ctochestlc iategr8lt € d(u):"*p - - r) , {lr\. "lJ, /{"oruu +ac(u)}ao where the function G(a) is monotone non-decreasingbounded continu- ous on the left with G(O\ : O (G is colled the Levy-Khinchin lvnction correspnding to X). Then f is X-integroble if and only if f belongsto the Orlicz spo,cetro[0,11, where o(c) : r/tr In the caseX is a p-stablemotion i.e. Q(u) : exp{-lt - sllulr} then tro[0,1]- trp[0,r] Vakhania and Kandelski [32] gave a definition of a stochastic in- tegral for operator-valued functions w.r.t. a Levy process taking values in a Hilbert space.[t and satisfyingEllx(l)ll'< *. Let.B denote. the corrariance operator associated with X. For any bounded linear operator Ae L(H,II) define ll,lll-: [t1,naa-)]'/' + [tr1,E-a,l)]t/'. Then the set {e : 1;a;;.: 0} is a linear semi-groupin the linear group L(H,I/). The function,a --+ll.all. is a norm in the correspondingfactor group. We shall not distinguish between a coset and the individual operatorsin the coset. Let A denotethe completionof L(H,I/) in this noim. Considerthe spacet rll1,l],1] of functions/ t [0,1] - I such that I f^ I lff(t)ll!ctt< x. J o I The standardextension procedure yields a stochasticintegral I f O dx(t) J o I f for each I e Lz[[o,t],I]. It was shownthat I f(t\dx(t) is a .E[- J o 6 DaurrgHung Thang valued infinitely divisible random variable with the characteristic func- tion given by I f F(h) : expI Klf. (t)(h)ldt, h e H, J o where K(h): lnG(h) and G(h) is the characteristicfunction of X(1). Rao [fS] used this stochastic integral to obtain the characteri- zation theorem for Wiener processestaking values in a Hilbert space .EI throught indentieal distributions A [ lplrdX(t) and I gQ)dx(t). JJ o Namely, he proved that under a slight assumption, I"'/(t) dX and I f I gU\ d,X arc indentically distributed if and only if Xis a ff-valued J o Wiener process and 1l tf I IQ)RI.Q) at : I s(t)Rs.(t) dt, JJ o0 where .B is the covarianceoperator associatedwith X. Here the integral is understood to be a Bochner integral under convergencein the spacel. 3. RANDOM MEASURES AND STOCHASTIC INTEGRALS WITH RESPECT TO THEM Let (S, S) be a measurable space. A mapping M from S into .Lo(O) is called a random measureif for every sequence{A"} of disjoint sets in S, the random variables M(A") are independent'and we have lrr(|/an):1rro", where the seriesis assumedto convergein probability. In addition, if M(A) is asymmetric random variable for every .4 € S then we call M a symmetric random measure. Thcory of ctocllstic intcgralr 7 By definition, a random measuneson (^9,S) can be viewed as a real stochastic process indexed by the parameter set S. Two random mea^suresM and N are called a version of each other if for each A € S P{w : M(A)(r) : N(A)(r)} : 1 For each o fixed, the set function A --+M(A) (or) is called a sample path of M. We say that a random measure is regular if its all paths are mea- sures of bounded variation.
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