Truncated Milstein Method And Modifications For Nonautonomous Stochastic Ordinary Differential Equations

In this thesis,the stochastic truncation Milstein method is used to approximate the true solution of non-autonomous stochastic differential equations and the convergence rate was proved to be min(1-2?,?)in the stochastic differential equations with super-linear state variables and H?lder continuous time variables,where ? is an arbitrarily small positive number and ? is the continuous exponent of the time variable H?der continuity.This means that the convergence rate may be affected by ?.When the smoothness of the time variable is very poor,that is,when ? < 1,the convergence order is ?;when ? > 1,the convergence order is very closed to 1,which is the same as the convergence rate of truncated Milstein in the case of autonomy.Moreover,in order to overcome the low convergence rate caused by the H?lder continuous time variable,we propose randomized truncated Milstein method,which applies the technique of randomization to the truncated Milstein method and applies it to non-autonomous stochastic differential equations,numerical examples are used to verify the accuracy of the theoretical conclusions.Finally,we use numerical examples to verify the accuracy of theoretical conclusions.