Euler-Maruyama Schemes For Multi-Term Fractional Stochastic Differential Equations

In recent decades,fractional differential equations have attracted people's increasing interest due to their wide applications in disciplines such as economics,physics,signal processing and control theory.Among them,multi-term fractional models are known as the potential mathematical tools to describe the complex systems and phenomena caused by different anomalous relaxations.However,it is very difficult to exactly solve the multi-term fractional differential equations.The analytical solutions of most equations cannot be obtained accurately,even if the analytical solutions of some equations can be obtained,they also need to be expressed by special functions which are difficult to be accurately calculated.Therefore,many works focus on the calculation of numerical solutions of multi-term fractional differential equations.In addition,it is well known that almost all mathematical models are influenced by noise factors.Thus,researchers in various fields pay more attention to a novel model that is fractional stochastic differential equations(FSDEs).In this thesis,we will construct and analyze an Euler-Maruyama(EM)method for the multi-term FSDEs with the initial value condition.And then the strong convergence of the proposed EM method is proved.Moreover,variable-order fractional stochastic differential equations have attracted more and more attention of scholars as a new research direction.Compared with the constant-order fractional stochastic differential equations,the variable-order fractional stochastic differential equation has variable order and its calculation is more complex.In the thesis,we extend the previous EM method to the numerical solution of variable-order fractional stochastic differential equations,and prove that the numerical solution also has strong convergence.The specific works of this thesis are organized as follows:In Chapter 2,the EM method for a class of multi-term Caputo fractional stochastic differential equations with the initial value condition is derived.Then,the strong convergence of the presented EM method is proved,and the convergence order is given.Thus,a fast implementation of the proposed EM method is also discussed by using the sum-ofexponentials approximation for the weakly singular kernel in the Riemann-Liouville fractional integral.Finally,three numerical examples are provided to illustrate the effectiveness of the EM method and verify the theoretical results.In Chapter 3,the previous EM method is extended to more general case in variable-order fractional stochastic differential equations.Firstly,it is proved that the variable-order FSDEs is well-posedness.Then,the strong convergence of the EM method is obtained and proved by an auxiliary equation.Finally,we carry out two numerical experiments to investigate the performance of the numerical method and to support the theoretical analysis.