Split-step Milstein methods for multi-channel stiff stochastic differential systems

Abstract

We consider split-step Milstein methods for the solution of stiff stochastic differential equations with an emphasis on systems driven by multi-channel noise. We show their strong order of convergence and investigate mean-square stability properties for different noise and drift structures. The stability matrices are established in a form convenient for analyzing their impact arising from different deterministic drift integrators. Numerical examples are provided to illustrate the effectiveness and reliability of these methods.

Double-implicit and split two-step Milstein schemes for stochastic differential equations

“…Zong et al [27][28][29] utilize split-step techniques to study linear theta Euler and Milstein methods. For more on this method, we refer to [6,19,25].…”

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

“…Stochastic ordinary di erential equations (SODEs) play a pivotal role in explaining some physical phenomena such as chemical reactions [9], nancial mathematics [10], mathematical ecology [11], epidemiology [12], medicine [13], and population dynamics [14]. Generally, SODEs cannot be solved analytical, but many numerical solutions can be found, for instance, the split-step theta Milstein method [15], the least-squares method [16], the discrete Temimi-Ansari method [17], the improved Euler-Maruyama method [18], the ve-stage Milstein method [19], the split-step Milstein method [20], the split-step Adams-Moulton Milstein method [21], the split-step forward Milstein method [22], and the Runge-Kutta method [23,24].…”

Truncated Milstein method for non-autonomous stochastic differential equations and its modification

“…The implicit methods of the Euler's type for SDEs were studied in [2,14,27,32,33]. The Milstein-type implicit methods for SDEs were discussed in [13,19,29,37]. The multi-stage implicit methods were investigated in [1,3,5].…”