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Wiener integrals, Malliavin calculus and covariance measure structure. (English) Zbl 1126.60046

The stochastic calculus, and in particular the construction of the Skorohod integral, are developed for a new class of processes. More precisely, one considers square integrable processes \(X\) having a “covariance structure measure”; this means that the covariance \(R(s,t)= \text{cov}(X_s,X_t)\) can be associated to a measure \(\mu\) on \([0,T]^2\). Gaussian processes are more particularly considered, in particular fractional Brownian motions, or more generally bifractional Brownian motions \[ R(s,t)=2^{-K}((t^{2H}+s^{2H})^K -| t-s| ^{2HK})\quad 0<H<1,\quad 0<K\leq1 \] in the case \(2HK\geq1\).
Malliavin derivation and Skorohod integration are developed in this setting. A more precise description is given in the Gaussian case. In particular, the relation with pathwise integrals is discussed, and an Itô formula is given.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60H05 Stochastic integrals
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