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

Abstract

In this article, 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 initial 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.

Keywords:

• Chaotic representation
• Divergence operator for Gaussian processes
• Fractional Brownian motion
• Fractional derivatives and integrals

Mathematics Subject Classification:

• 60H20
• 60H07

Acknowledgments

The authors thank the Nucleus Millennium Information and Randomness P01-005 for support. Partially supported by CONACyT grant 45684-F and by the program Millennium Nucleus in Information and Randomness.