Research On Stability And Numerical Algorithms Of Neutral Stochastic Delay Differential Equations

Neutral stochastic delay differential equations(NSDDEs)can be regarded as a generalization of stochastic delay differential equations(SDDEs).The mathematical models described by these equations not only consider the interference of random factors,but also consider the influence of the past growth rate of variables on the current state and are applied to finance,biology,machinery and other fields.So it is of great importance to study the numerical solutions of these equations.Firstly,for constructing the numerical solutions of NSDDEs,this paper uses a modified implicit Milstein(MIM)algorithm based on the fully implicit EM algorithm and the Milstein algorithm,and it overcomes the problem that the Milstein algorithm needs to use unknown initial values when solving NSDDEs.At the same time,the MIM algorithm is a fully implicit numerical scheme that can be used to solve stiff problems.However,the MIM algorithm contains complex derivation,which will greatly increase the amount of calculation in numerical experiments.Aiming at the problem,on the basis of MIM algorithm,this paper proposes a modified semi-implicit derivative-free(MSIDF)algorithm without derivative.Secondly,this paper studies the use of fully implicit EM algorithm and MIM algorithm to solve NSDDEs.Not only obtain the conditions such that numerical solutions of NSDDEs using the fully implicit EM algorithm are almost surely exponentially stable,but also obtain the numerical decay rate.At the same time,the corresponding conditions are also given for the MIM algorithm so that the numerical solutions of NSDDEs obtained by the algorithm are almost surely exponentially stable.Furthermore,this paper also studies the use of MSIDF algorithm to solve linear and nonlinear NSDDEs,and obtains the conditions and numerical decay rate that make the numerical solutions almost surely exponentially stable.In particular,this paper also considers the semi-implicit EM algorithm to solve linear NSDDEs,and more specific conditions and numerical decay rates such that the numerical solutions almost surely exponentially stable are obtained.Finally,this paper gives three specific NSDDEs,namely two one-dimensional nonlinear NSDDEs and one two-dimensional linear NSDDE.They are solved using the fully implicit EM algorithm,MIM algorithm and MSIDF algorithm.For the implicit equations appearing in the calculation,this paper uses Newton's iterative algorithm to solve.And all algorithms are implemented in MATLAB.Through numerical experiments,the correctness of the theoretical results and the effectiveness of the method are verified.