Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion
Abstract
Description
As a general rule, differential equations driven by a multi-dimensional irregular path $Γ$ are solved by constructing a rough path over $Γ$. 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 $α< 1/2$. 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 $α\in (0, 1)$ may be solved on the closed upper halfplane, and that the solutions have finite variance.