The Annals of Probability

On probability laws of solutions to differential systems driven by a fractional Brownian motion

F. Baudoin, E. Nualart, C. Ouyang, and S. Tindel

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Abstract

This article investigates several properties related to densities of solutions $(X_{t})_{t\in[0,1]}$ to differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/4$. We first determine conditions for strict positivity of the density of $X_{t}$. Then we obtain some exponential bounds for this density when the diffusion coefficient satisfies an elliptic type condition. Finally, still in the elliptic case, we derive some bounds on the hitting probabilities of sets by fractional differential systems in terms of Newtonian capacities.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2554-2590.

Dates
Received: January 2014
Revised: February 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139149

Digital Object Identifier
doi:10.1214/15-AOP1028

Mathematical Reviews number (MathSciNet)
MR3531675

Zentralblatt MATH identifier
1352.60081

Subjects
Primary: 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

Keywords
Fractional Brownian motion rough paths Malliavin calculus hitting probability

Citation

Baudoin, F.; Nualart, E.; Ouyang, C.; Tindel, S. On probability laws of solutions to differential systems driven by a fractional Brownian motion. Ann. Probab. 44 (2016), no. 4, 2554--2590. doi:10.1214/15-AOP1028. https://projecteuclid.org/euclid.aop/1470139149


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