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

Jeremie Unterberger

doi:10.1214/ecp.v15-1574

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 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 $\alpha\in(0,1)$ may be solved on the closed upper half-plane, and that the solutions have finite variance.

Highlights

Assume Γt = (Γt(1), . . . , Γt(d)) is a smooth d-dimensional path, and V1, . . . , Vd : Rr → Rr be smooth vector fields
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 ∈ (0, 1) may be solved on the closed upper halfplane, and that the solutions have finite variance
A stochastic calculus for multidimensional fractional Brownian motion with arbitrary Hurst index

Summary

Introduction

Is the difficulty of getting estimates for the iterated integrals Γ; another reason lies in the essence of the rough path method which relies on pathwise estimates; a third reason is, that the Chen series diverges even in the simplest cases (onedimensional usual Brownian motion for instance) as soon as the vector fields are unbounded and non-linear, e.g. quadratic. We prove convergence of the series (0.3) when the vector fields Vi are linear and Γ is analytic fBm (afBm for short) This process – first defined in [8] –, depending on an index α ∈ (0, 1), is a complex-valued process, a.s. κ-Holder for every κ < α, which has an analytic continuation to the upper half-plane Π+ := {z = x + iy | x ∈ R, y > 0}. Constants (possibly depending on α) are generally denoted by C, C′, C1, cα and so on

Definition of afBm and first estimates

Estimates for iterated integrals of Γ

Proof of main theorem