Milstein-type semi-implicit split-step numerical methods for nonlinear stochastic differential equations with locally Lipschitz drift terms

Abstract

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of non-linear stochastic differential equations with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein, and truncated Milstein procedures on non-linear stochastic differential equations including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

Untitled

“…Thus, determinant operator G 2 of linear system equatıon (14), (15) has bounded inverse G −1 2 . Therefore solution of linear system equatıon ( 14), ( 15) is defined by…”

“…For a detailed description of the integral boundary conditions, we refer the reader to a recent paper [8]. For more details of nonlocal and integral boundary conditions, see [9][10][11][12][13][14][15][16] and references therein.…”

“…The order of convergence of the Euler method for neutral stochastic functional differential equations has been studied in [12] where also similar order of convergence has been achieved. Convergence performance of different numerical integrators have been discussed in [13,14] specifically for the Langevin-type equations, and weak convergence of order one has been obtained. [15] considered the same Langevin-type equation…”

“…Many researchers has been studied extensively the well-posedness of nonlocal boundary value problems of elliptic type partial differential equations in the Euclidean space, which is a flat manifold, (see, e.g. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein).…”

Some results for the weak convergence of semi-implicit split-step methods

“…In 2018, they also developed and investigated the high dimensional versions of SISS methods for nonlinear SDEs with non-Lipschitz drift terms in [12]. After that, in [13], they presented Milstein type semi-implicit split step (MSISS) methods for nonlinear SDE with locally Lipschitz drift terms. They also work on strong convergence analysis of the SISS and MSISS methods in [14] and [15], respectively.…”