Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift

Abstract

In this note we propose an exact simulation algorithm for the solution of dX_t=dW_t+b(X_t)dt (1) where b is a smooth real function except at point 0 where b(0+)\neq b(0-). The main idea is to sample an exact skeleton of X using an algorithm deduced from the convergence of the solutions of the skew perturbed equation dX^\beta_t=dW_t+b(X^\beta_t)dt + \beta dL^0_t {X^\beta} (2) towards X solution of (1) as \beta tends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in \cite{etoremartinez1} for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as \beta tends to 0, we prove that this limit algorithm is an exact simulation algorithm for the solution of the limit equation (1). Numerical examples are shown to illustrate the performance of this exact simulation algorithm.