Convergence Rates of Split-Step Theta Methods for SDEs with Non-Globally Lipschitz Diffusion Coefficients

Abstract

The present work analyzes the mean-square approximation error of split-step theta methods in a non-globally Lipschitz regime. We show that under a coupled monotonicity condition and polynomial growth conditions, the considered methods with the parameters $θ ∈ [1/2, 1]$ have convergence rate of order $1/2.$ This covers a class of stochastic differential equations with super-linearly growing diffusion coefficients such as the popular $3/2$-model in finance. Numerical examples support the theoretical results.