Stability Of Numerical Scheme For Hybrid Stochastic Differential Equations

Stochastic differential equations (SDEs) with Markov switching can or Hybrid SDE be used to describe physical dynamics under environment change, and the finan-cial models fluctuating between various economic circumstances. The introduction of Markov chain into the SDEs illustrates the switching among different financial models. For instance, the stock market has different characteristics in bear market and bull market, making it difficult to describe the behavior using a simple model of stochastic differential equations. In this case, the SDEs with Markov switching is an excellent way to describe the market behaviors. Therefore, SDEs with Markov switching are taking an increasingly important part in modeling the complicated financial markets.Generally, hybrid stochastic differential equations can not be solved analytically and hence numerical methods must be used. Such as Euler-Maruyama methods, Mil-stein methods, explicit order 1.0 strong scheme and so on. The numerical stability is very important characteristic of numerical methods, since an unstable numerical method may result in malignant growth of rounding error and distraction of the nu-merical solution, therefore studying on the numerical stability of stochastic differential equations is a very important issue.In this paper, we first study the stability regions for Euler-Maruyama approxima-tions of hybrid stochastic differential equations. Then, the pth moment exponential stability of Milstein scheme is discussed. For scalar linear multiplied-noise equations, this paper proves that the Milstein scheme approximation is pth moment exponential stable if the stepsize is small enough. And for mean square exponential stability, in par-ticular, the existence conditions showed in this paper are weaker than the in literature. Meanwhile, we make a further study of higher order approximations:explicit order 1.0 strong scheme. The result is that there also exists an interval of stepsize where the pth moment exponential stability of explicit order 1.0 strong scheme approximations.The sufficient conditions for stability of time-varying hybrid stochastic differential equations are given as well.