Pathwise convergence of a linearization scheme for stochastic differential-algebraic equations under the local Lipschitz coefficients

TL;DR

This paper introduces a local linearization numerical scheme for index-1 stochastic differential-algebraic equations with nonlinear coefficients, proving pathwise convergence and demonstrating efficiency in high-dimensional problems.

Contribution

The paper develops a new linearization-based numerical method for SDAEs with singular matrices, proving its convergence and efficiency in high dimensions.

Findings

The method converges in the pathwise sense with rate 1/2 minus epsilon.

Implementation verifies the theoretical convergence rate.

The approach effectively handles high-dimensional SDAEs with local Lipschitz coefficients.

Abstract

The paper deals with the numerical treatment of index-1 stochastic differential-algebraic equations (SDAEs) with nonlinear coefficients that satisfy the local Lipschitz and the Khasminskii conditions. The key challenge here is the presence of a singular and non-autonomous matrix in the equation, which makes the numerical method challenging to analyze. To tackle this challenge, we develop a more general numerical method using a local linearization technique. More precisely, we use the Taylor expansion to decompose locally the drift component of the SDAEs in linear and nonlinear parts. The linear part is approximated implicitly and must resolve the singularity issue of each time step, while the nonlinear part is approximated explicitly. This method is fascinating due to the fact that it is efficient in high dimension. We prove that this novel numerical method converges in the pathwise sense with rate $\frac{1}{2}-\epsilon$, for arbitrary $\epsilon >0$. The implementation of this novel numerical method is also carried out to verify our theoretical result.