Some Numerical Methods For Solving Several Classes Of Stochastic Differential Equations

Stochastic differential equations can describe mathematical and physical processes disturbed by random factors or processes with uncertainty,so stochastic differential equation models widely exist in social production and scientific research.It becomes fairly necessary to develop effective numerical methods and perform numerical simulations,since most of stochastic differential equations can not be solved exactly.Numerical methods are often required to maintain the unique structure of the original system when they are designed.Symplectic methods for stochastic Hamiltonian systems play an important role in the structure-preserving algorithms for stochastic differential equations.Due to the complexity of solving implicit numerical methods,estimating possible higher order partial derivatives and solving tedious symplectic conditions as well as order conditions,most of the stochastic symplectic methods have low computational efficiency and only few of them have a high convergence order.Moreover,a large number of stochastic differential equations in real life do not satisfy the Lipschitz condition and the linear growth condition,however,the researches on the strong convergence of the high-order numerical methods for stochastic differential equations under the local Lipschitz conditions and super-linear growth conditions are still scarce.In this thesis,some studies have been conducted on numerical methods for several types of stochastic Hamiltonian systems and stochastic differential equations under local Lipschitz conditions and super-linear growth conditions.The main work of this thesis is as follows:An autonomous Stratonovich type stochastic Hamiltonian system is considered.Based on the theory of generating functions with noises,a new class of stochastic symplectic methods containing at most the first partial derivatives of the coefficient functions is constructed.It extends the stochastic symplectic Runge-Kutta methods.The mean-square convergence order conditions of the methods,which are simplified according to the relationship of coefficients of the colored rooted tree,are analyzed by using the colored rooted trees theory.Finally,concrete numerical methods of order 1.0 for solving noncommutative and commutative stochastic Hamiltonian systems are constructed,respectively.A non-autonomous Stratonovich type stochastic Hamiltonian system with additive noises is considered.A alass of simplified stochastic partitioned Runge-Kutta methods is proposed and its mean-square convergence order conditions and symplectic conditions are given by using the colored rooted tree theory.Then stochastic symplectic partitioned Runge-Kutta methods of mean-square order 1.5 are constructed.Besides,the symplectic and order conditions are simplified for the separable stochastic Hamiltonian system and the second-order stochastic Hamiltonian system,as a result,several classes of explicit stochastic symplectic partitioned Runge-Kutta methods are constructed.A non-autonomous Stratonovich type stochastic Hamiltonian system is considered.Sufficient conditions of stochastic Runge-Kutta methods to maintain the symplectic property are given by using the colored rooted tree theory.In the autonomous case,these conditions are proved to be equivalent with the coefficient-type symplectic conditions of stochastic Runge-Kutta methods.The theoretical results are applied to the construction of the pseudo-symplectic stochastic Runge-Kutta methods and several explicit stochastic Runge-Kutta methods of high pseudo-symplectic order are constructed for stochastic Hamiltonian systems with additive noises.An It (?) type stochastic differential equation with non-global Lipschitz and superlinear growth coefficients is considered.This thesis combines the projection strategy with the explicit It (?)-Taylor method and proposes the projected explicit It (?)-Taylor method as well as a selection strategy of projected parameter.The stochastic C-stability and stochastic B-compatibility of the method under non-global Lipschitz conditions and super-linear growth conditions,which further proves the mean-square convergence of the method,are analyzed.In each part of this work,numerical examples are carried out.The numerical results fully verify the validity of the constructed numerical method and the correctness of the theoretical results.