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

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 θ\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 θ\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 θ\in(1/2, 1], these two theta-Milstein schemes can share the exponential mean-square stability of the exact solution. For θ\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.