Abstract
The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.
Highlights
Many real-world phenomena in different fields of science, such as biology, financial engineering, neural network, and wireless communications, can be simulated by the Itostochastic differential equations (SDEs) of the form dy (t) = f (t, y (t)) dt + g (t, y (t)) dW (t), t > 0, (1)y (t0) = y0, where f(t, y(t)) is the drift coefficient and g(t, y(t)) is the diffusion coefficient and the Wiener process W(t) is defined on a given probability space (Ω, F, P) with a filtration {Ft}t≥0 which satisfied the usual conditions
If we look closely to the stability conditions of the split-step methods which are based on Milstein method to approximate the diffusion part in SDEs (2), we can find that the term (1/2)y2 plays an important role in the stability
We are interested in mean-square (MS) stability of the drifting split-step theta Milstein (DSSθM) methods for stochastic differential equations (SDEs)
Summary
The necessary and sufficient conditions for their MS stability are given, and stability regions and A-stable approaches have been discussed These methods improved the convergence order to be 1.0, we can see that the MS stability conditions of these methods have some restriction on the parameters and step-size. With restriction on step-size, the authors proved that under one-sided Lipschitz condition the two classes of theta Milstein methods can share the exponential MS stability of the exact solution for nonlinear SDEs. Nowadays, split-step numerical methods have attracted a lot of attention from scholars for SDEs and have been proven to be a very efficient approach. We discussed the MS stability regions and A-stable approaches of DSSθM methods for linear SDEs. we proved that, under local Lipschitz condition, the numerical methods can preserve the asymptomatic MS stability of the nonlinear SDEs without restriction on step-size.
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