Strong convergence rates of stochastic theta methods for index 1 stochastic differential algebraic equations under non-globally Lipschitz conditions

Abstract

This work investigates numerical approximations of index 1 stochastic differential algebraic equations (SDAEs) with non-constant singular matrices under non-global Lipschitz conditions. Analyzing the strong convergence rates of numerical solutions in this setting is highly nontrivial, due to both the singularity of the constraint matrix and the superlinear growth of the coefficients. To address these challenges, we develop an approach for establishing mean square convergence rates of numerical methods for SDAEs under global monotonicity conditions. Specifically, we prove that each stochastic theta method with $\theta \in [\frac{1}{2},1]$ achieves a mean square convergence rate of order $\frac{1}{2}$. Theoretical findings are further validated through a series of numerical experiments.