Some cases of chaotic representation. (Quelques cas de représentation chaotique.)(French)Zbl 0754.60043

Séminaire de probabilités, Lect. Notes Math. 1485, 10-23 (1991).
[For the entire collection see Zbl 0733.00018.]
The author considers a martingale $$X$$ with respect to a filtration $$({\mathcal F}_ t)$$ such that $$\langle X,X\rangle_ t=t$$; $$({\mathcal N}^ X_ t)$$ is the corresponding natural filtration of $$X$$. Let $$S$$ be the disjoint union of all $$S_ n$$, $$n\geq 0$$, where $$S_ n\subset(0,\infty)^ n$$ 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 $$L^ 2(S,\lambda)$$ to $$L^ 2$$; let $$H(X)$$ be its image. A more general stochastic integral $$\int\chi_{{\mathcal A}_ T} fdX$$ is defined, for every $$f\in L^ 2({\mathcal B}(S)\otimes{\mathcal F}_ T)$$ null outside $${\mathcal A}_ T=\{(A,\omega); A\subset(T(\omega),\infty)\}$$, where $$T$$ is a stopping time; in its definition $$(X_{T+t})$$ is used. Let $$H^ T(X)$$ be its image. If $${\mathcal N}^ X_ 0$$ is trivial and for every $$({\mathcal N}^ X_ t)$$-martingale $$M_ t$$ there is an $${\mathcal N}^ X$$-previsible $$\phi$$ with $$dM_ t=\phi_ t dX_ t$$, then $$X$$ is said to have PRP (propriété de représentation prévisible), while, if $$H(X)$$ is the whole $$L^ 2({\mathcal N}^ X_ \infty)$$, $$X$$ is said to have PRC (... chaotique).
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 $$Z_ t=X_ t$$ for $$t\leq T$$, $$Z_ t=X_ T+Y_{t-T}-Y_ 0$$ for $$t\geq T$$, where $$X$$, $$Y$$ are independent, both having PRC, and $$T$$ is an $$({\mathcal N}^ X_ t)$$-stopping time. The second is a $$Y$$ for which there exist $$X^ n$$ having PRC and $$({\mathcal N}^{X^ n}_ t)$$-stopping times $$T_ n$$ such that $$Y=X^ n$$ on $$[0,T_ n]$$ and $$\sup T_ n=\infty$$. The third is an $$X$$ having PRP, with $$\langle X,X\rangle_ t=t$$, $$d[X,X]_ t=dt+\phi_ t dX_ t$$, $$\phi$$ being previsible, nowhere null, with $$\int \chi_{[0,t]}(s)\phi^{-2}_ s ds<\infty$$ for all $$t$$. The fourth is $$X$$ from the solution $$(X,E)$$ (its existence and unicity in law are shown) of $$d[X,X]_ t=dt+dE_ t$$, $$dE_ t=E_{t-}\lambda dX_ t$$, $$X_ 0=x$$, $$E_ 0=e$$, where $$\lambda$$, $$x$$, $$e$$ are constants. The proofs begin by “Proposition 1”, relative to two martingales $$X$$, $$Y$$ with $$\langle X,X\rangle_ t=\langle Y,Y\rangle_ t=t$$ 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 $$({\mathcal N}^ X_ t)$$-stopping time. Proposition 1 involves also $$g=C_ T(U,X)$$, $$h=C(U,X)$$ for $$U\in L^ 2$$, where the projections of $$U$$ on $$H^ T(X)$$ and $$H(X)$$ are $$\int \chi_{{\mathcal A}_ T}gdX$$, $$\int hdX$$, respectively. The paper finishes with other two results. The first expresses $$C(U,X)$$ using $$C(V,X)$$’s with $${\mathcal F}_ T$$-measurable $$V$$’s and $$C_ T(U,X)$$ and the second proves that a sufficient condition for $$U\in H(X)$$ is that $$X$$ has PRP and $$\int E(C_{\inf A-}(A)^ 2)\lambda(dA)<\infty$$, where $$dC_ t(A)=\Gamma_ t(A)dX_ t$$ and, for $$A=\{\dots<b<c\}$$, $$C_ t(A)$$ is $$E(U;{\mathcal F}_ t)$$ for $$t\geq c$$, and $$E(\Gamma_ c(A);{\mathcal F}_ t)$$ for $$t\in[b,c)$$ etc.

MSC:

 60G44 Martingales with continuous parameter 60H05 Stochastic integrals

Zbl 0733.00018
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