Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations

TL;DR

This paper analyzes the convergence and stability of split-step and stochastic theta-Milstein schemes for SDEs with non-global Lipschitz coefficients, establishing strong convergence orders and stability preservation under various theta parameters.

Contribution

It provides new theoretical results on the convergence order and stability properties of two classes of theta-Milstein schemes for SDEs with non-global Lipschitz coefficients.

Findings

Strong convergence order 1 for   with  ; under linear growth, convergence order is standard.

Schemes preserve exponential mean-square stability for  ; under linear growth, stability is also preserved.

Results apply to SDEs with non-global Lipschitz coefficients, broadening applicability.

Abstract

This paper examines convergence and stability of the two classes of theta-Milstein schemes for stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients: the split-step theta-Milstein (SSTM) scheme and the stochastic theta-Milstein (STM) scheme. For \theta\in[1/2,1], this paper concludes that the two classes of theta-Milstein schemes converge strongly to the exact solution with the order 1. For \theta \in [0,1/2], under the additional linear growth condition for the drift coefficient, these two classes of the theta-Milstein schemes are also strongly convergent with the standard order. This paper also investigates exponential mean-square stability of these two classes of the theta-Milstein schemes. For \theta\in(1/2, 1], these two theta-Milstein schemes can share the exponential mean-square stability of the exact solution. For \theta\in[0, 1/2], similar to the convergence, under the additional linear growth condition, these two theta-Milstein schemes can also reproduce the exponential mean-square stability of the exact solution.