Stochastic Differential Equations with Additive Fractional Noise and Locally Unbounded Drift

Abstract

Let $$\{ B_{t}^{{H,}}t \in [0,T]\}$$ be a fractional Brownian motion with Hurst parameter $$H < \tfrac{1}{2}$$ . We prove the existence and uniqueness of a strong solution for a stochastic differential equation of the form $${{X}_{t}} = {{x}_{0}} + B_{t}^{H} + \smallint _{0}^{t}b(s,{{X}_{s}})ds$$ , where b(s, x) is not locally bounded and satisfies a suitable integrability condition.

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