Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. (English) Zbl 1169.60013

The following stochastic differential equation on \(\mathbb{R}^d\): \[ X^i_t=x_0^i+\sum^m_{j=1}\int^t_0\sigma^{ij}(X_s)d B^j_s+\int^t_0 b^j(X_s)ds,\quad t\in[0,T],\quad i=1,\dots,d,\tag{1} \] where \(x_0\in\mathbb{R}^d\) is the initial value of the process \(X\) and \(B= \{B_t,t\geq 0\}\) is an \(m\)-dimensional fractional Brownian motion of Hurst parameter \(H\in(\tfrac 12,1)\), is considered. The authors study the regularity of the solution to equation (1) in the sense of Malliavin calculus. They prove the differentiability of the solution in the directions of the Cameron-Martin space and the absolute continuity with respect to the Lebesgue measure of the solution under ellipticity condition on the coefficient \(\sigma\).


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
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