Convergence of a positivity preserving logarithmic truncated EM method for SDEs with discontinuous drift coefficients

Abstract

In this paper, a class of nonlinear stochastic differential equations with positive solutions and discontinuous drift coefficients is studied, considering both theoretical and computational aspects. The theoretical results focus on the existence of a unique positive solution for such SDEs via the approach introduced by Müller-Gronbach et al. (2022), and the computational aspect utilises the truncated Euler-Maruyama method proposed by Li et al. (2023) together with a logarithmic transformation that ensures a positive approximation to the original solution. The convergence of the numerical method is investigated, and the boundedness of the $p$-th moment is obtained. Finally, the proposed method is used to verify the convergence with the help of some numerical examples.