Convergence and Stability of Numerical Solution of Stochastic Differential Equations with Piecewise Continuous Arguments 

Author  DaiHongYu 
Tutor  LiuMingZhu 
School  Harbin Institute of Technology 
Course  Computational Mathematics 
Keywords  Differential equations with piecewise continuous arguments Convergence Mean square asymptotically stable Numerical method 
CLC  O211.63 
Type  Master's thesis 
Year  2008 
Downloads  91 
Quotes  0 
Differential equations with piecewise continuous arguments(EPCA) as important models are applied widely in many fields, such as physical, biological systemsand control theory. This equation had attracted much attention, and many useful conclusion was obtained. However, up to now there is no one who has considered theinfluence of noise. Actually, the environment and accidental events may greatly influence 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 stability 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. TheEulerMaruyama 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 EulerMaruymamanumerical 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 semiimplicit Eulermethod and Milstein method are studied. For the semiimplicit 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 semiimplicit 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.