Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion

Jérémie Unterberger (1)

1 Probabilités et statistiques

As a general rule, differential equations driven by a multi-dimensional irregular path $\Gamma$ are solved by constructing a rough path over $\Gamma$. The domain of definition – and also estimates – of the solutions depend on upper bounds for the rough path; these general, deterministic estimates are too crude to apply e.g. to the solutions of stochastic differential equations with linear coefficients driven by a Gaussian process with Hölder regularity $\alpha < 1/2$. We prove here (by showing convergence of Chen's series) that linear stochastic differential equations driven by analytic fractional Brownian motion [7, 8] with arbitrary Hurst index $\alpha \in (0, 1)$ may be solved on the closed upper halfplane, and that the solutions have finite variance.

Mots clés: stochastic differential equations; frectional Brownian motion; analytic fractional brownian motion; rough paths; Hölder continuity; Chen series

Domaines: Probabilités [math.PR]

Jérémie Unterberger. Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion. Electronic Communications in Probability, 2010, 15, pp.411-417. ⟨hal-00381655⟩