Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition

Abstract

The present article revisits the well-known stochastic theta methods (STMs) for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. Under a coupled monotonicity condition in a domain $D \subset {{\mathbb {R}}}^d, d \in {{\mathbb {N}}}$, we propose a novel approach to achieve upper mean-square error bounds for STMs with the method parameters $\theta \in [\tfrac{1}{2}, 1]$, which only get involved with the exact solution processes. This enables us to easily recover mean-square convergence rates of the considered schemes, without requiring a priori high-order moment estimates of numerical approximations. As applications of the error bounds, we derive mean-square convergence rates of STMs for SDEs driven by three kinds of noises under further globally polynomial growth condition. In particular, the error bounds are utilized to analyze approximation of SDEs with small noise. It is shown that the stochastic trapezoid formula gives better convergence performance than the other STMs. Furthermore, we apply STMs to the Ait-Sahalia-type interest rate model taking values in the domain $D = ( 0, \infty )$, and successfully identify a convergence rate of order one-half for STMs with $\theta \in [\tfrac{1}{2}, 1]$, even in a general critical case. This fills the gap left by Szpruch et al. (BIT Numer Math 51(2):405–425, 2011), where strong convergence of the backward Euler method was proved, without revealing a rate of convergence, for the model in a non-critical case.