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Stochastic calculus with respect to Gaussian processes. (English) Zbl 1015.60047
The authors consider a family of Gaussian processes \((B_t)_{t\in {\mathbb R}_+}\) of the form \(B_t = \int_0^t K(t,s) dW_s\), where \(K\) is a deterministic kernel and \((W_t)_{t\in {\mathbb R}_+}\) is a standard Wiener process. They construct a stochastic calculus with respect to such processes via the stochastic calculus of variations, using the anticipating Skorokhod integral operator with respect to \((W_t)_{t\in {\mathbb R}_+}\), which is denoted by \(\delta\). The stochastic integral of an adapted process \(u\) with respect to \((B_t)_{t\in {\mathbb R}}\) is defined to be \(\delta (K^* u)\), where \(K^*\) is the adjoint of the operator with kernel \(K\). Itô and Stratonovich change of variable formulas and Hölder regularity results are proved for indefinite integrals with respect to \((B_t)_{t\in {\mathbb R}}\), for a wide class of deterministic (singular and regular) kernels \(K\). The results apply in particular to fractional Brownian motion with Hurst parameter \(H\in (1/4,1/2)\).

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
Full Text: DOI
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