Stability Of Analytical Solution And Numerical Solution For Stochastic Differential Equations With Piecewise Continuous Arguments

Stochastic differential equations with piecewise continuous arguments(SEPCAs)is a special form of stochastic delay differential equations. As an importantmathematical model, It has been widely used in many fields such as biology, signalprocessing and control theory. There are theoretical significance and real applicationvalue to study the property of solution forSEPCAs.Stability is a very important property in the stochastic differential equationtheory, it reflects the influence of the initial value, the perturbation of the coefficientand parameters for the solution of equation. Therefore, it is necessary to study thestability of the solutions forSEPCAs. In this paper, we study the mean-squarestability of the equilibrium and numerical solutions of the equation.First, we study the mean-square stability of the equilibrium and numericalsolution of one-dimensional SEPCA. we give the form of existing analyticalsolutions and the necessary and sufficient conditions of asymptotic mean-squarestability of the analytical solution. we obtain numerical solution of the equation byθ-method, The sufficient and necessary conditions of asymptotical mean-squarestability of the numerical solution are given.Next, we study asymptotic mean-square stability of the equilibrium andnumerical solutions of the multi-dimensionalSEPCA. we based on the vectorisationof matrices and Kronecker product, we transform the mean-square stability problemfor equilibrium solution of the equation into stability problem for a deterministicequation. The sufficient and necessary conditions of asymptotic mean-squarestability of the equilibrium solution are given.Last, we obtain the numerical solution of the multi-dimensional SEPCAbyθ-method. By using logarithmic norm, we give the sufficient conditions ofasymptotic mean-square stability of the numerical solution.