Balanced numerical schemes for SDEs with non-Lipschitz coefficients

Abstract

In this chapter, we discuss numerical methods for SDEs with coefficients of polynomial growth. The nonlinear growth of the coefficients induces instabilities, especially when the nonlinear growth is polynomial or even exponential. For stochastic differential equations (SDEs) with coefficients of polynomial growth at infinity and satisfying a one-sided Lipschitz condition, we prove a fundamental mean-square convergence theorem on the strong convergence order of a stable numerical scheme in Chapter 5.2. We apply the theorem to a number of existing numerical schemes. We present in Chapter 5.3 a special balanced scheme, which is explicit and of half-order mean-square convergence. Some numerical results are presented in Chapter 5.4. We summarize the chapter in Chapter 5.5 and present some bibliographic notes on numerical schemes for nonlinear SODEs. Three exercises are presented for interested readers.