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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 + of the form B t = 0 t K(t,s)dW s , where K is a deterministic kernel and (W t ) t + 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 + , which is denoted by δ. The stochastic integral of an adapted process u with respect to (B t ) t is defined to be δ(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 , for a wide class of deterministic (singular and regular) kernels K. The results apply in particular to fractional Brownian motion with Hurst parameter H(1/4,1/2).

MSC:
60H05Stochastic integrals
60H07Stochastic calculus of variations and the Malliavin calculus
60G15Gaussian processes