Convergence and Stability of Numerical Solution of Stochastic Differential Equations with Piecewise Continuous Arguments
|School||Harbin Institute of Technology|
|Keywords||Differential equations with piecewise continuous arguments Conver-gence Mean square asymptotically stable Numerical method|
Differential equations with piecewise continuous arguments(EPCA) as impor-tant models are applied widely in many fields, such as physical, biological systemsand control theory. This equation had attracted much attention, and many useful con-clusion was obtained. However, up to now there is no one who has considered theinfluence of noise. Actually, the environment and accidental events may greatly in-fluence the systems. Thus analyzing stochastic differential equations with piecewisecontinuous arguments(SEPCA) is an interesting topics both in theory and applications.The paper focus on the stability of the analytical solution, convergence and sta-bility of numerical solutions for SEPCA.The paper surveys the recent developments of the study of stochastic differentialequations(SDEs) and EPCA from the point of existence, uniqueness, stability of theanalytical solution and the convergence and stability of numerical methods.For a special class of linear SEPCA, the sufficient and necessary condition whichguarantees the mean square asymptotic stability of the zero solution is given. TheEuler-Maruyama method is defined and the mean square asymptotic stability of thenumerical solution is discussed. Under the condition which guarantees the meansquare asymptotic stability of the zero solution, it is proved that the Euler-Maruymamanumerical solutions are mean square asymptotically stable if the stepsize satisfiessome restrictions. In addition, the condition of the restriction is given.For a general linear test equation, a sufficient condition which guarantees themean square asymptotic stability of the zero solution is obtained. The convergencein mean square sense and mean square asymptotic stability of the semi-implicit Eulermethod and Milstein method are studied. For the semi-implicit Euler method, severalimportant inequalities related to the analytical solution are proved and the local orderof the numerical solutions in mean square sense is obtained accordingly. Further,it is proved that the semi-implicit Euler method is convergent of order 12 in meansquare sense for the linear test equation by virtue of properties of the conditionalexpectation. For Milstein method, it is convergent with order 1 in mean square sense.By studying the difference equation, the conditions which guarantee the numerical methods to be mean square asymptotically stable are given. In addition, the conditionof the restriction of h are given.Moreover, the relevant numerical experiments are given in every part. On onehand, these experiments verify the results obtained in theory, on the other hand, theinfluences of the stepsize and parameters of the numerical methods on the stability areillustrated visually.