Numerical Methods For Stochastic Differential Equations With Markovian Switching

Stochastic system with Markovian switching is an important class of stochastic hybrid systems and has more wider applications in science and engineering fields. So it has received a great deal of attention. However, in general, the explicit solutions are not known.We therefore consider computable discrete approximations. The convergence and stability of numerical methods are two of key problems in numerical analysis. Convergence and stability of numerical methods are well understood for stochastic differential equations with uniform Lipschitz continuous coefficients. In 2011, Higham and Hu demonstrated that a backward Euler-Maruyama method strongly converges to the solution of non-linear the stochastic differential equations with one-sided Lipschitz drift and linearly growing diffusion coefficients. In 2013, Mao and Szpruch demonstrated that the stochastic theta method strongly converges to the solution to non-linear stochastic differential equation with onesided linearly growing condition and one-sided Lipuschitz drift coefficients. This paper uses the similar methods to that of Mao and Szpruch's paper to demonstrate the convergence and stability of the backward Euler-Maruyama method for stochastic differential equations with Markovian switching with one-sided Lipuschitz and one-sided linearly growing coefficients.This paper investigates the existence,uniqueness,stability and boundness of the solution with the aid of stochastic analysis, such as the relevant properties of stochastic integrals,It ?o lemma,stopping theory, Burkholder-Davis-Gundy inequality, Gronwall inequality and so on. Then, we prove the strong convergence of the backward Euler-Maruyama method on the basis of several lemmas and investigate the stability of the numerical approximation with the semi-martingle convergence theorem. Last, we demonstrate the convergence and stability of stochastic theta approximation for stochastic differential equations.