Revistes Catalanes amb Accés Obert (RACO)

Intégrales stochastiques de processus anticipants et projections duales prévisibles

C. Donati-Martin, M. Yor


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.

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