Convergence rates of theta-method for neutral SDDEs under non-globally Lipschitz continuous coefficients

TL;DR

This paper analyzes the convergence rates of theta-method schemes for neutral stochastic differential delay equations with highly nonlinear coefficients, under non-globally Lipschitz conditions, driven by Brownian motion and jumps.

Contribution

It provides new convergence rate results for theta-EM schemes applied to complex neutral SDDEs with non-globally Lipschitz coefficients.

Findings

Established strong convergence rates for theta-EM schemes

Proved almost sure convergence under specified conditions

Extended analysis to equations driven by jumps

Abstract

This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of $\theta$-EM schemes are given for these equations driven by Brownian motion and pure jumps respectively, where the drift terms satisfy locally one-sided Lipschitz conditions, and diffusion coefficients obey locally Lipschitz conditions, and the corresponding coefficients are highly nonlinear with respect to the delay terms.