The truncated $\theta$-Milstein method for nonautonomous and highly nonlinear stochastic differential delay equations

TL;DR

This paper introduces a truncated θ-Milstein numerical method for nonautonomous stochastic differential delay equations with polynomially growing coefficients, demonstrating near-one convergence rate under weaker assumptions than traditional monotonicity, supported by numerical validation.

Contribution

The paper develops a truncated θ-Milstein method with proven strong convergence for complex stochastic delay equations, relaxing previous monotonicity constraints.

Findings

Convergence rate close to one established

Method effective for polynomial growth coefficients

Numerical example confirms theoretical results

Abstract

This paper focuses on the strong convergence of the truncated $\theta$-Milstein method for a class of nonautonomous stochastic differential delay equations whose drift and diffusion coefficients can grow polynomially. The convergence rate, which is close to one, is given under the weaker assumption than the monotone condition. To verify our theoretical findings, we present a numerical example.