Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

Abstract

Abstract In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters $$H&lt;\frac{1}{2}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.