A note on the asymptotic stability of the Semi-Discrete method for Stochastic Differential Equations

2020/08/06 by Nikolaos Halidias, Halidias, Nikolaos, Ioannis S. Stamatiou +1
Economics, Econometrics and Finance · #60H10 #60H35 #65C20 #65C30 #65J15 #65L20 #FOS: Mathematics #Numerical Analysis (math.NA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2008.03148

openalex publication_date 2020/08/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of \mathcal L²-convergence of the truncated SD method and showed that it can be arbitrarily close to 1/2, see Stamatiou, Halidias (2019), Convergence rates of the Semi-Discrete method for stochastic differential equations, Theory of Stochastic Processes, 24(40). We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE, which we call Lamperti semi-discrete (LSD). Numerical simulations support our theoretical findings.

Related

• Convergence rates of the Semi-Discrete method for stochastic differential equations
• Lamperti Semi-Discrete method
• A note on pathwise stability and positivity of nonlinear stochastic differential equations
• On the numerical solution of some nonlinear stochastic differential\n equations using the semi-discrete method
• A note on convergence and stability of the truncated Milstein method for stochastic differential equations