Convergence And Stability Of Numerical Solutions To SDDEs With Markovian Switching

This dissertation mainly studies SSTE and SLTE methods for stochastic differ-sntial delay equations with Markovian switching and obtains the convergence and stability of numerical solutions. First of all, we study the existence and uniqueness of the analytical solutions, stability and convergence of the numerical methods for SDEs and SDDEs. Meanwhile,we introduce some traditional applications and research on S-DDEs. At the basic theory part of this paper, we give some basic theorems, definitions and notations which will be used in next chapters. Next,we research the convergence and stability of numerical solutions.If f satisfies the polynomial Lipschitz as well as its first variate satisfies the one-side Lipschitz and its second variate satisfies global Lipschitz, g satisfies global Lipschit2 condition, it is proved that:(1)Under ??[0,1/2], SSTE and SLTE schemes converge strongly to the exact solution with order 1/2 if f satisfies an additional linear growth condition; (2)Under ??(1/2,1], both SSTE and SLTE schemes are convergent with order 1/2.Under a coupled condition, which is used to prove exponential mean square stabil-ity of the exact solution, we have proved that(1) for ?? [0,1/2], under the condition that f satisfies an additional linear growth condition, SSTE and SLTE schemes are exponentially mean square stable with a step-size restriction; (2) for ?? (1/2,1], both SSTE and SLTE schemes are exponentially mean square stable. In fact, these two methods are not only exponentially mean square stable under the corresponding con-ditions, but also the decay rate as measured by the bound of the Lyapunov exponent can be reproduced arbitrarily.On the last part of this paper,we give numerical experiment to support the con-clusion that reflect the influences of the step-size on stability of numerical methods.This dissertation aim at stochastic differential delay equations with Markovian switching and we prove the convergence and stability of numerical solutions. We have improved the conclusions in the current literature.