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Exact simulation for solutions of one-dimensional StochasticDifferential Equations with discontinuous drift
Published online by Cambridge University Press: 22 October 2014
Abstract
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In this note we propose an exact simulation algorithm for the solution of (1)\begin{equation} \label{eds-intro} {\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t,\quad X_0=x, \end{equation}where \hbox{$\bar{b}$}
b̅ is a smooth real function except at point0 where \hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}
b̅(0 + ) ≠b̅(0 −). The main idea is to sample an exact skeleton ofX using analgorithm deduced from the convergence of the solutions of the skew perturbed equation(2)\begin{equation} \label{edsbeta} {\rm d}X^\beta_t={\rm d}W_t+\bar{b}(X^\beta_t){\rm d}t + \beta {\rm d}L^0_t(X^\beta),\quad X_0=x \end{equation}towards X solution of (1) as β ≠0 tends to 0. In this note, we show that this convergence induces the convergenceof exact simulation algorithms proposed by the authors in [Pierre Étoré and MiguelMartinez. Monte Carlo Methods Appl. 19 (2013) 41–71] for thesolutions 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 anexact simulation algorithm for the solution of the limit equation (1). Numerical examples are shown to illustratethe performance of this exact simulation algorithm. 
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- © EDP Sciences, SMAI 2014
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