An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz Coefficients

Abstract

We consider the approximation of one-dimensional stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler--Maruyama discretization scheme that allows us to prove strong convergence, with a rate. Under some regularity and integrability conditions, we obtain the optimal strong error rate. We apply this scheme to SDEs widely used in the mathematical finance literature, including the Cox--Ingersoll--Ross (CIR), the 3/2, and the Ait-Sahalia models, as well as to a family of mean-reverting processes with locally smooth coefficients. We numerically illustrate the strong convergence of the scheme and demonstrate its efficiency in a multilevel Monte Carlo setting.