An explicit approximation for super-linear stochastic functional differential equations

TL;DR

This paper introduces an explicit truncated Euler-Maruyama scheme for super-linear stochastic functional differential equations, overcoming the limitations of implicit methods and ensuring stability and convergence without requiring global Lipschitz conditions.

Contribution

It develops a novel explicit numerical method for super-linear SFDEs that relaxes the global Lipschitz restriction and guarantees stability and convergence.

Findings

The scheme is bounded and converges in L^p.

Convergence rate is 1/2 order.

Numerical solutions preserve exponential stability.

Abstract

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler-Maruyama scheme for super-linear SFDEs, and obtain the boundedness and convergence in L^p. We also yield the convergence rate with 1/2 order. Different from some previous works, we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory.