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Stochastic calculus with respect to Gaussian processes. (English) Zbl 1015.60047
The authors consider a family of Gaussian processes ${\left({B}_{t}\right)}_{t\in {ℝ}_{+}}$ of the form ${B}_{t}={\int }_{0}^{t}K\left(t,s\right)d{W}_{s}$, where $K$ is a deterministic kernel and ${\left({W}_{t}\right)}_{t\in {ℝ}_{+}}$ 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 ${\left({W}_{t}\right)}_{t\in {ℝ}_{+}}$, which is denoted by $\delta$. The stochastic integral of an adapted process $u$ with respect to ${\left({B}_{t}\right)}_{t\in ℝ}$ is defined to be $\delta \left({K}^{*}u\right)$, 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 ${\left({B}_{t}\right)}_{t\in ℝ}$, for a wide class of deterministic (singular and regular) kernels $K$. The results apply in particular to fractional Brownian motion with Hurst parameter $H\in \left(1/4,1/2\right)$.

##### MSC:
 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60G15 Gaussian processes