Niveau: Supérieur, Doctorat, Bac+8
MALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS JORGE A. LEON AND SAMY TINDEL Abstract. In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Holder 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 C∞-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. 1. Introduction The recent progresses in the analysis of differential equations driven by a fractional Brownian motion, using either the complete formalism of the rough path analysis [3, 10, 18], or the simpler Young integration setting [25, 33], allow to study some of the basic properties of the processes defined as solutions to rough or fractional equations. This global program has already been started as far as moments estimates [13], large deviations [16], or properties of the law [2, 21] are concerned. It is also natural to consider some of the natural generalizations of diffusion processes, arising in physical applications, and see if these equations have a counterpart in the fractional Brownian setting.
- algebraic structure
- then
- stochastic differential
- brownian motion
- a1 ?
- young integration
- differential equations driven
- rn-valued random variable
- standard brownian