Convergence And Stability Of Numerical Methods For Several Classes Of Stochastic Differential Equations

Stochastic differential equations(SDEs)play a significant part in many application fields,such as economy,finance and automatic control.In recent years,the research on the theory and applications of SDEs has attracted a lot of attention.However,many SDEs often possess super-linear grownth coefficients in practice and such SDEs do not have any explicit solutions in general.Consequently,efficient numerical methods are important for the applications of SDEs and constructing appropriate numerical schemes for solving this kind of nonlinear SDEs is of great,both theoretical and practical,interest.This thesis mainly investigates the convergence and stability of numerical methods for several classes of SDEs,the whole thesis contains the following aspects.(1)Under the non-global Lipschitz condition,the strong convergence and stability of the truncated Euler-Maruyama(EM)methods for SDEs with Poisson jumps are studied.When the drift and diffusion coefficients satisfy the super-linear growth condition and the jump coefficient satisfies the linear growth condition,the L~r(r?2)-convergence result for the truncated EM methods is established.When all the three coefficients of SDEs are allowing to grow super-linearly,the L~r(0<r<2)-convergence order are also investigated.Moreover,it is proved that the truncated EM method preserves nicely the mean square exponential stability and asymptotic boundedness of the underlying SDEs with Poisson jumps.(2)Under the non-global Lipschitz condition,the strong convergence and stability of the tamed EM methods for neutral stochastic differential delay equations(NSDDEs) are investigated.Two types of explicit tamed EM schemes are constructed for the NSDDEs with coefficients of superlinearly growth.The first type is convergent in the L~r(r?2)sense under the local Lipschitz plus Khasminskii-type conditions.The second type is of order one half in the mean-square sense under the Khasminskii-type,global monotonicity and polynomial growth conditions.Moreover,it is proved that the partially tamed EM scheme has the property of mean-square exponential stability.(3)A generalized Ait-Sahalia model with Poisson jumps is proposed.The analytical properties including positivity,boundedness and pathwise asymptotic estimations of the solution to this model are investigated by the method of Lyapunov function.Moreover,it is proved that the EM numerical solution converges to the true solution of the model in probability.Finally,the effectiveness and accuracy of the methods are analyzed by a numerical example.(4)By using the approximate value at the nearest grid-point on the left of the delayed argument to estimate the delay function,a class of split-step?-method for solving the stochastic delay age-dependent population equations with Markovian switching is proposed.It is shown that the numerical method is convergent under the given conditions.Numerical examples are provided to illustrate our results.(5)Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion(G-SDDE)and the EM method are studied.Under the global Lipschitz condition,it is shown that the G-SDDE is exponentially stable in mean square if and only if for sufficiently small step size,the EM method is exponentially stable in mean square.Thus,numerical simulations based on the EM method can be used to investigate the exponential stability of the underlying G-SDDE in the absence of an appropriate Lyapunov function.(6)In the G-framework,strong convergence of the numerical methods for the neutral stochastic differential equation with time-dependent delay is examined.For the numerical simulation,the EM scheme is proposed and is proved to be convergent with order one half under the global Lipschitz condition.Compared with the previ-ous literature on the SDEs driven by classical Brownian motion,neutral term and time-dependent delay as well as the G-Brownian motion noise are considered in our scheme for wilder practical applications.