EXACT SIMULATION FOR SOLUTIONS OF ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS DRIFT

In this note we propose an exact simulation algorithm for the solution of dX(t) - dW(t) + (b) over bar (X-t)d(t), X-0 - x, where (b) over bar is a smooth real function except at point 0 where (b) over bar (0+) not equal (b) over bar (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(t)(beta) = dW(t) + (b) over bar (X-t(beta))dt + beta dL(t)(0)(X-beta), X-0 = x towards X solution of (1) as beta not equal 0 tends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in [Pierre Etore and Miguel Martinez. Monte Carlo Methods Appl. 19 (2013) 41-71] 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.