MOMENT ESTIMATES FOR SOLUTIONS OF LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY ANALYTIC FRACTIONAL BROWNIAN MOTION

As a general rule, differential equations driven by a multi-dimensional irregular path are solved by constructing a rough path over. 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 Holder 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 [6, 7] with arbitrary Hurst index alpha is an element of (0, 1) may be solved on the closed upper half-plane, and that the solutions have finite variance.