Almost Sure Exponential Stability Of Numerical Methods For Stochastic Delay Differential Equations

As stochastic delay differential equations (SDDEs) established, this class ofequations has been received a great deal of attention and has been applied in manyfields, such as medicine, economics, automatic control etc. Since few differentialequations can be solved explicitly, it is necessary to develop the numerical methodsfor these equations. In view of the importance of stability theory in numericalanalysis, this paper mainly investigats the stability of two numerical methods forSDDEs, which are non-Euler type. This paper is arranged as follows:In Chaper1, the history of SDDEs as well as numerical stability theory ofnumerical methods for SDDEs is reviewed briefly. The main contents are introducedin general.In Chapter2, some basic notations, definitions, assumptions and useful lemmasare introduced as prelimineries of this paper.In Chapter3, the conditions under which Stochastic Split-step Theta (SST)method can reproduce the almost sure exponential stability of the exact solutions tostochastic delay differential equations (SDDEs) for any [0,1]are investigated.The numerical experiment confirms the correctness of the theorem obtained.In Chaper4, one new scheme named Stochastic Non-Standard Finite Difference(SNSFD) method for a special class of SDDEs is introduced, which can reproducethe almost sure exponential stability of the exact solutions of the SDDEs as stated inTheorem4.2. Theorem4.3gives the corresponding result in n-dimension.Numerical simulations show the effectiveness of the conclusions.The meaning of this paper is to study the stability of two non-Euler typenumerical methods. And by the theorems obtained before, the conditions underwhich the methods can reproduce the almost sure exponential stability of the exactsolutions of SDDEs are given.