Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift. (22nd October 2014) - Ã‰torÃ©, Pierre; Martinez, Miguel
<x content-type="archive" xml:space="preserve">Abstract</x> In this note we propose an exact simulation algorithm for the solution of (1) <alternatives> <![CDATA[\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}]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxs5j" xmlns:xlink="http://www.w3.org/1999/xlink" />dXt=dWt+bÌ…(Xt)dt, â€X0=x, </alternatives> where <alternatives> <![CDATA[\hbox{$\bar{b}$}]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsb4" xmlns:xlink="http://www.w3.org/1999/xlink" /><textual-form>bÌ…</textual-form></alternatives> is a smooth real function except at point 0 where <alternatives> <![CDATA[\hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxshq" xmlns:xlink="http://www.w3.org/1999/xlink" /><textual-form>bÌ…(0 + ) â‰  bÌ…(0 âˆ')</textual-form></alternatives>. 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 (2) <alternatives> <![CDATA[\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}]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxs83" xmlns:xlink="http://www.w3.org/1999/xlink" />dXtÎ²=dWt+bÌ…(XtÎ²)dt+Î²dLt0(XÎ²), â€X0=x</alternatives> towards X solution of (1) as Î² â‰  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 Ã‰torÃ© and Miguel Martinez. Monte Carlo Methods Appl. 19 (2013) 41â€"71] for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as Î² 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.