Intégrales stochastiques de processus anticipants et projections duales prévisibles

We define a stochastic anticipating integral $\delta^\mu$ with respect to Brownian motion, associated to a non adapted increasing process $(\mu_t)$, with dual projection $t$. The integral $\delta^\mu (u) $ of an anticipating process $(u_t)$ satisfies: for every bounded predictable process $f_t$, $$...

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Bibliographic Details
Published in:Publicacions matemàtiques 1999, Vol.43 (1), p.281-301
Main Authors: Donati-Martin, C., Yor, M.
Format: Article
Language:eng ; fre
Subjects:
Ecuaciones diferenciales estocásticas
Integración estocástica
Movimiento browniano
Proceso de difusión
Procesos estocásticos
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Summary:We define a stochastic anticipating integral $\delta^\mu$ with respect to Brownian motion, associated to a non adapted increasing process $(\mu_t)$, with dual projection $t$. The integral $\delta^\mu (u) $ of an anticipating process $(u_t)$ satisfies: for every bounded predictable process $f_t$, $$ E\left[\left(\int f_s\, dB_s\right) \delta^\mu (u)\right ] = E\left[ \int f_s u_s \, d\mu_s\right]. $$ We characterize this integral when $\mu_t = \sup_{t \leq s \leq 1} B_s$. The proof relies on a path decomposition of Brownian motion up to time 1.
ISSN:0214-1493
2014-4350
DOI:10.5565/PUBLMAT_43199_13