Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

David Nualart (1) , Bruno Saussereau (2)

1 Department of mathematics
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H > 0:5. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition

Mots clés: Stochastic differential equation; Malliavin calculus; Fractional Brownian motion

Domaines: Probabilités [math.PR]

David Nualart, Bruno Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Processes and their Applications, 2009, 119 (2), pp.391-409. ⟨10.1016/j.spa.2008.02.016⟩. ⟨hal-00485648⟩