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

2013/01/14 by Pierre Étoré, Pierre Etore, Etore, Pierre +3
Economics, Econometrics and Finance · Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1301.3019

openalex publication_date 2013/01/14 · arxiv created 2013/10/04 · arxiv updated 2013/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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+)≠ 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^β_t=dW_t+b(X^β_t)dt + βdL⁰_t X^β (2) towards X solution of (1) as βtends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in \citeetoremartinez1 for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as βtends 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.

Related

• Exact Simulation of One-dimensional Stochastic Differential Equations involving the local time at zero of the unknown process
• Numerical Algorithms for 1-d Backward Stochastic Differential Equations: Convergence and Simulations
• Unbiased simulation of stochastic differential equations
• An adaptive strong order 1 method for SDEs with discontinuous drift coefficient
• Skew-symmetric schemes for stochastic differential equations with non-Lipschitz drift: an unadjusted Barker algorithm