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

TL;DR

This paper demonstrates that linear stochastic differential equations driven by analytic fractional Brownian motion with any Hurst index can be solved on the upper halfplane, with solutions having finite variance, extending rough path theory to this setting.

Contribution

It proves convergence of Chen's series for solutions driven by analytic fractional Brownian motion, enabling solutions with finite variance for all Hurst indices.

Findings

Solutions exist on the upper halfplane.

Solutions have finite variance.

Convergence of Chen's series established.

Abstract

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 H\"older 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.