Explicit Numerical Method Construction And Asymptotic Property Approximation For Several Classes Of Non-Linear Stochastic Systems

Super-linear stochastic systems with special structures,including fast-slow stochastic differential equations（SFSDEs）with time-scale separation,distribution-dependent stochastic differential equations（DDSDEs）（also called McKean-Vlasov stochastic or mean-field stochastic differential equations）and distribution-dependent stochastic delay differential equations（DDSDDEs）,are widely used as mathematical models in physics,chemistry,biology,finance,and other application fields.Due to the complex structure of these three types of stochastic differential equations（SDEs）,their explicit solutions are usually unavailable,which poses significant challenges for theoretical analysis and practical applications.Therefore,it is an urgent problem to develop easy-to-apply and efficient numerical methods for these three types of systems and approximate their asymptotic properties.Given the advantages of the explicit method,such as simple form,clear structure and easy operation,this paper focuses on constructing explicit numerical methods for these three types of non-linear stochastic systems and approximating their asymptotic properties.For a class of nonlinear SFSDEs,constructing an explicit multiscale truncated EM scheme and analyzing its convergence.Under the condition that the slow equation coefficients are locally Lipschitz continuous with respect to the slow variable,by combining the heterogeneous multi-scale method,truncation idea,and the result of the averaging principle,the multi-scale truncated Euler-Maruyama（EM）scheme is constructed for SFSDEs,and the strong convergence between the numerical solution and the exact solution of the slow variables is proved.Furthermore,under the condition that the slow equation coefficients have polynomial growth,the strong convergence rate between the numerical solution and the exact solution of the averaged equation is provided.Furthermore,using the strong convergence rate of the averaging principle can further lead to a strong convergence rate between the numerical solution and the exact solution of the slow variable of SFSDEs.Finally,the theoretical results are validated through numerical experiments.For a class of non-linear DDSDEs,constructing the truncated EM methods and analyzing the asymptotic properties of the numerical solutions.First,the propagation of chaos theory in L^p（p≥2）is developed for DDSDEs,possibly with both drift and diffusion coefficients having super-linear growth in the state variable.Then,by the stochastic particle method,the truncated EM method is proposed for this class of DDSDEs.The convergence between the numerical and exact solutions is analyzed in the L^p（p≥2）sense,and a 1/2-order optimal convergence rate in time step size is obtained.Furthermore,it is proven that the numerical solution generated by the truncated EM scheme can inherit the long-time dynamic properties of DDSDEs,including moment uniform boundedness,exponential stability,and stability in distribution.Furthermore,the convergence of the numerical invariant measure and the underlying invariant measure of DDSDEs in the W₂-Wasserstein metric is also proved.Finally,some numerical experiments are provided to verify the theoretical results.Investigating the approximation theory of the invariant measures of DDSDEs using the EM method.Based on the assumption of the existence and uniqueness of the invariant measure for DDSDEs,a self-interacting process that depends only on the historical information of the solution is constructed for DDSDEs.The convergence rate of the weighted empirical measure of the self-interacting process and the invariant measure of DDSDEs is obtained in the W₂-Wasserstein metric.Furthermore,under the condition of linear growth,an EM scheme is constructed for the self-interacting process,and a uniformly 1/2-order convergence rate with respect to time is obtained.Based on this convergence rate,the convergence rate between the weighted empirical measure of the EM numerical solution of the selfinteracting process and the invariant measure of DDSDEs is derived.Moreover,an appropriate multi-particle system is constructed,and the convergence rate between the averaged weighted empirical measure of the EM numerical solution of the multiparticle system and the invariant measure of DDSDEs in the W₂-Wasserstein metric is also given.Moreover,the computational cost of the two approximation methods is compared.It is shown that the averaged weighted empirical approximation of the particle system has a lower cost.Finally,the theoretical results are validated through numerical experiments.For a class of non-linear DDNSDDEs,as a particular class of DDSDDEs,establishing the well-posedness of a strong solution and constructing an explicit tame EM method.Firstly,the well-posedness of strong solutions for non-linear DDNSDDEs is proven by using the technique of distribution iteration.Secondly,the propagation of chaos is established for DDNSDDEs.By utilizing the stochastic particle method,the tamed EM scheme is constructed for the interacting particle system,and an optimal 1/2-order convergence rate between the numerical and the exact solutions of the particle system is obtained.By virtue of the propagation of chaos,the convergence rate between the numerical and the exact solutions of DDNSDDEs is further provided.Finally,the theoretical results are validated through a numerical example.