Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Simple Search

Query:
Enter a query and click »Search«...
Format:
Display: entries per page entries
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

Login Username: Password:

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences
Published by Springer-Verlag | Webmaster