Convergence And Stability Of Numerical Methods For Stochastic Differential Equations With Piecewise Continuous Arguments

Stochastic differential equations with piecewise continuous arguments(SDEPCAs)and SDEPCAs driven by Poisson random measure are widely applied in economics,biology,control theory,neural network and other fields.Since,in general,it is difficult to obtain the exact solution to an SDEPCA or SDEPCA driven by Poisson random measure,numerical approximations are required in practice.And hence,it is of great theoretical and practical significance to study the properties of numerical methods.This thesis deals with the convergence and stability of numerical solutions for SDEPCAs and SDEPCAs driven by Poisson random measure.For SDEPCAs with global Lipschitz continuous coefficients,the convergence and stability of one-leg θ methods in small moment(p ∈(0,1))are investigated.Firstly,under global Lipschitz conditions,the pth moment convergence of one-leg θ methods is analysed.And then,based on the convergence,the equivalent relation between the pth moment exponential stability of SDEPCAs and that of one-leg θ methods is investigated.For SDEPCAs in which the drift coefficients are polynomially growing and the diffusion coefficients satisfy global Lipschitz conditions,since the explicit Euler method is divergent,an explicit tamed Euler method is constructed.The pth(p ≥ 1)moment boundedness of numerical solutions is considered.In addition,the pth moment convergence and convergence order are investigated.When both the drift coefficients and diffusion coefficients do not satisfy the global Lipschitz condition,the strong convergence and stability of the implicit split-step θ(SST)method are studied.At first,under the local Lipschitz condition,the monotone condition and the one-sided Lipschitz condition,the pth(p ∈ [1,2))moment convergence of the SST method with θ ∈ [1/2,1] is considered.After that,the sufficient conditions which make SDEPCAs exponentially stable in mean square sense are analysed.Moreover,the sufficient conditions under which the SST method reproduces the mean square exponential stability are discussed.The pth(p ≥ 2)moment convergence of SST methods is analysed for some kinds of SDEPCAs in which both drift and diffusion coefficients are polynomially growing with respect to delay terms.The sufficient conditions under which the order of SST methods reaches to 1/2 are discussed.At the same time,the stability of the improved split-step theta(ISST)method is discussed.As an extension of SST methods for SDEPCAs,the convergence and stability of the compensated split-step θ(CSST)method are investigated for SDEPCAs driven by Poisson random measure.The sufficient conditions under which the CSST method is pth(p ≥ 2)moment convergent are analysed.The order is also considered.For different ranges of θ,the sufficient conditions under which the CSST method reproduces the exponential mean square stability of the underlying system are dicussed with θ ∈ [0,1/2] and θ ∈(1/2,1],respectively.