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Zbl 0754.60043
Emery, M.
Some cases of chaotic representation. (Quelques cas de représentation chaotique.) (French)
[A] Séminaire de probabilités, Lect. Notes Math. 1485, 10-23 (1991).

[For the entire collection see Zbl 0733.00018.]\par The author considers a martingale $X$ with respect to a filtration $({\cal F}\sb t)$ such that $\langle X,X\rangle\sb t=t$; $({\cal N}\sp X\sb t)$ is the corresponding natural filtration of $X$. Let $S$ be the disjoint union of all $S\sb n$, $n\ge 0$, where $S\sb n\subset(0,\infty)\sp n$ is the set of all $n$-element subsets of $(0,\infty)$, with their elements written in an ascending order, and let $\lambda$ be the direct sum of the corresponding Lebesgue measures. There is a linear isometric mapping $f\to\int fdX$ from $L\sp 2(S,\lambda)$ to $L\sp 2$; let $H(X)$ be its image. A more general stochastic integral $\int\chi\sb{{\cal A}\sb T} fdX$ is defined, for every $f\in L\sp 2({\cal B}(S)\otimes{\cal F}\sb T)$ null outside ${\cal A}\sb T=\{(A,\omega); A\subset(T(\omega),\infty)\}$, where $T$ is a stopping time; in its definition $(X\sb{T+t})$ is used. Let $H\sp T(X)$ be its image. If ${\cal N}\sp X\sb 0$ is trivial and for every $({\cal N}\sp X\sb t)$-martingale $M\sb t$ there is an ${\cal N}\sp X$-previsible $\phi$ with $dM\sb t=\phi\sb t dX\sb t$, then $X$ is said to have PRP (propriété de représentation prévisible), while, if $H(X)$ is the whole $L\sp 2({\cal N}\sp X\sb \infty)$, $X$ is said to have PRC (... chaotique).\par The main purpose of this paper is to give new examples of $X$'s having PRC (there are 7 quotations with such examples). The first is $Z\sb t=X\sb t$ for $t\le T$, $Z\sb t=X\sb T+Y\sb{t-T}-Y\sb 0$ for $t\ge T$, where $X$, $Y$ are independent, both having PRC, and $T$ is an $({\cal N}\sp X\sb t)$-stopping time. The second is a $Y$ for which there exist $X\sp n$ having PRC and $({\cal N}\sp{X\sp n}\sb t)$-stopping times $T\sb n$ such that $Y=X\sp n$ on $[0,T\sb n]$ and $\sup T\sb n=\infty$. The third is an $X$ having PRP, with $\langle X,X\rangle\sb t=t$, $d[X,X]\sb t=dt+\phi\sb t dX\sb t$, $\phi$ being previsible, nowhere null, with $\int \chi\sb{[0,t]}(s)\phi\sp{-2}\sb s ds<\infty$ for all $t$. The fourth is $X$ from the solution $(X,E)$ (its existence and unicity in law are shown) of $d[X,X]\sb t=dt+dE\sb t$, $dE\sb t=E\sb{t-}\lambda dX\sb t$, $X\sb 0=x$, $E\sb 0=e$, where $\lambda$, $x$, $e$ are constants. The proofs begin by ``Proposition 1'', relative to two martingales $X$, $Y$ with $\langle X,X\rangle\sb t=\langle Y,Y\rangle\sb t=t$ and $X=Y$ on $[0,T]$, $T$ being a stopping time. In the last statement of this proposition, $X$ is PRC and $T$ is an $({\cal N}\sp X\sb t)$-stopping time. Proposition 1 involves also $g=C\sb T(U,X)$, $h=C(U,X)$ for $U\in L\sp 2$, where the projections of $U$ on $H\sp T(X)$ and $H(X)$ are $\int \chi\sb{{\cal A}\sb T}gdX$, $\int hdX$, respectively. The paper finishes with other two results. The first expresses $C(U,X)$ using $C(V,X)$'s with ${\cal F}\sb T$-measurable $V$'s and $C\sb T(U,X)$ and the second proves that a sufficient condition for $U\in H(X)$ is that $X$ has PRP and $\int E(C\sb{\inf A-}(A)\sp 2)\lambda(dA)<\infty$, where $dC\sb t(A)=\Gamma\sb t(A)dX\sb t$ and, for $A=\{\dots<b<c\}$, $C\sb t(A)$ is $E(U;{\cal F}\sb t)$ for $t\ge c$, and $E(\Gamma\sb c(A);{\cal F}\sb t)$ for $t\in[b,c)$ etc.
[I.Cuculescu (Bucureşti)]

MSC 2000:: *60G44 Martingales with continuous parameter
60H05 Stochastic integrals

Keywords: property of chaotic representation; property of previsible representation; recollement; martingale; stochastic integral; existence and unicity in law

Citations: Zbl 0733.00018

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