Numerical Metheods For Two Classes Of Stochastic Delay Differential Equations With Markovian Switching

Stochastic delay differential equations as an important mathematical model are ap-plied widely in many scientific fields, for example, finance, biology, medicine, environ-mental, demographic, control etc. As the explicit solutions can be hardly obtained, con-structing appropriate numerical methods and discussing the properties of the numericalsolutions are very essential in application and theory.In this paper,we study two kinds of numerical methods which are applied to stochas-tic delay differential equations with Markovian swithcing, the covergence and stability ofnumerical solutions are discussed .At first ,we focus on the existence and uiqueness of the analytical solutions, cov-ergence and stability of the numerical methods for stochastic differential equations andstochastic differential delay equations.Meanwhile, development history and the currentresearch situation are presented.By the introduction on the basic theory of stochastic delay differential equations , wegive some basic theorems , definitions, and notation which will be used in next chapters.The convergence and stability in the square sense of Milstein method for Fokker-Planck equation witn Markovian switching is considered. It is proved that the Milsteinmethod is convergent with order 1/2 under the Lipschitz and linear growth conditons.Inadditon, the conditons of stability and stepsize restriction are obtained.Next,we research the convergence and stability in the square sense of Euler-maruyama method applied to stochastic pantograph equations with Markovian switching.By Burkholder-Davis-Gundy inequality,Gronwall inequality ,elmentary inequality ,weget convergence of the numerical solutions with order 1/2 ,the conditions of the stabilityin the square sense of the numerical solutions and the stepsize restriction.After the discussion of convergence and stability for the numerical methods, we givenumerical experiments to support the conclusion which also re?ect the in?uences of thestep-size on stability of the numerical methods .All results given in this paper are new, so the work is valuable.Keywords: Stochastic delay differential equation with Markovian switching, Milstein...