A numerical scheme for stochastic differential equations with distributional drift
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In this paper we present a scheme for the numerical solution of stochastic differential equations (SDEs) with distributional drift. The approximating process, obtained by the scheme, converges in law to the (virtual) solution of the SDE in a general multi-dimensional setting. When we restrict our attention to the case of a one-dimensional SDE we also obtain a rate of convergence in a suitable L^1-norm. Moreover, we implement our method in the one-dimensional case, when the drift is obtained as the distributional derivative of a sample path of a fractional Brownian motion. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift cannot be expressed as a function of the state.
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