Convergence And Stability Of Exponential Euler Method For Stochastic Delay Differential Equations

Stochastic delay differential equations as a kind of important mathematicsmodels, have considered the effect of random factor and delay factor, which hastruly represented the objective facts. So far, stochastic delay differential equationshas been applied in the areas of biology, physics, chemistry, economics biomedicineetc. As a result, it becomes more and more necessary to have more studies onstochastic differential delay equations.Because it is quite difficult to get the solution of stochastic delay differentialequations, sometime even you can get the expression of analytical solution, whichcannot be used to study the quality of some system as its complexity. Therefore, theconstruction of numerical methods play an important role. Over the past fewdececades, there have been more and more studies on the stochastic delaydifferential equations and more relative conclusion have been made. However,compared with the ordinary differential equations, the study on stochastic differ-ential delay equations still at the state of infancy. Therefore, in this paper, theexponential Euler method will be applied in the stochastic differential delayequations so as to construct its data pattern and then make a discussion aboutconvergence and stability the exponential Euler method.This paper lists several common numerical methods and Ruge-Kutta method inordinary differential equations to obtain its first-order form, namely the exponentialEuler method. The first main part proves that the convergence order is0.5, andgives a numerical example given to verify the convergence order, The second majorcontent is analysis of the stability of the exponential Euler method. The paper firstresearches the application of exponential Euler method to linear stochastic delaydifferential equation, then describes a sufficient condition for exponential stabilityin mean square of exponential Euler method, and lists two sets of numericalexperiments. From the experiment, we found that ensuring its stability, the rangeof step length of the exponential Euler method is larger than that of Euler method.Then the paper considers semi-linear stochastic delay differential equations, andworks out a sufficient condition for the exponential Euler method’s exponentialstability in mean square.