## 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 $$\sigma$$.

### MSC:

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

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