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

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

2009-05-06

arxiv.org/abs/0905.0782v1

Mathematics

Probability

Scientific paper

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.

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