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

¹ ENSIMAG – Laboratoire Jean Kuntzmann, Tour IRMA 51, rue des Mathématiques, 38041 Grenoble cedex 9, France
pierre.etore@imag.fr
² Université Paris-Est Marne-La-Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, 5 Bld Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
miguel.martinez@univ-mlv.fr

Received: 11 February 2013
Revised: 3 June 2013

Abstract

In this note we propose an exact simulation algorithm for the solution of $\mathrm{d}{\mathit{X}}_{\mathit{t}}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}\mathrm{\right)}\mathrm{d}\mathit{t,} {\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x,}$(1)where is a smooth real function except at point 0 where . 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 $\mathrm{d}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{\right)}\mathrm{d}\mathit{t}\mathrm{+}\mathit{\beta }\mathrm{d}{\mathit{L}}_{\mathit{t}}^{\mathrm{0}}\mathrm{\left(}{\mathit{X}}^{\mathit{\beta }}\mathrm{\right)}\mathit{,} {\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x}$(2)towards X solution of (1) as β ≠ 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 Étoré 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 β 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.