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**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.

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.

Reviewer: I.Cuculescu (Bucureşti)