Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

Abstract

In this paper we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here the coefficients are deterministic, the inital condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results. IntroductionThe fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1) is a Gaussian process with useful properties. In particular, the stationary of its increments, and the self-similarity and the long-range dependence of this process (see Mandelbrot and Van Ness [13]) become the fBm a suitable driven noise for the construction of stochastic models and the analysis of phenomena that exhibit scale-invariant and long-range correlated force. However the fBm is not a semimartingale when H = 1 2 . Hence we cannot apply the techniques of the stochastic calculus in the Itô sense to define a stochastic integral with respect to the fBm.Different interpretations of stochastic integral with respect to the fBm B have been used by several authors to study the fractional stochastic differential equation of the form(1.1)In the case that H ∈ (1/2, 1), it is reasonable to consider equation (1.1) as a path-by-path ordinary differential equation since B has Hölder-continuous paths with all exponents less than H and T 0 Y s dB s exists as a pathwise Riemann-Stieltjes integral for any λ-Hölder continuous process Y with λ > 1 − H (see Young [22]). This pathwise equation has been studied by several authors (see for instance [7, 8, 11, 19] and [23]). Zähle [23] has improved this pathwise approach * Partially supported by CONACyT

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