Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions

Fabrice Baudoin (1) , Cheng Ouyang (1) , Samy Tindel (2, 3)

1 Department of Mathematics
2 IECN - Institut Élie Cartan de Nancy
3 BIGS - Biology, genetics and statistics

In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H> 1/3. We show that under some geometric conditions, in the regular case H>1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H>1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

Domaines: Probabilités [math.PR]

Fabrice Baudoin, Cheng Ouyang, Samy Tindel. Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions. 2011. ⟨hal-00639187⟩