Malliavin calculus for fractional delay equations

Jorge A. Leon (1) , Samy Tindel (2)

1 CINVESTAV-IPN - Departamento de Control Automático
2 BIGS - Biology, genetics and statistics

In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a smooth density. To this purpose, we use Malliavin calculus based on the Frechet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.

Mots clés: Delay equation; Young integration; Fractional Brownian motion; Malliavin calculus

Domaines: Probabilités [math.PR]

Jorge A. Leon, Samy Tindel. Malliavin calculus for fractional delay equations. Journal of Theoretical Probability, 2012, 25 (3), pp.854-889. ⟨10.1007/s10959-011-0349-4⟩. ⟨hal-00440655⟩