Convergence Analysis Of Split-step Collocation Method For Linear Stochastic Volterra Integral Equations

From the botanist Brown found the Brownian motion in 1827, it was found in real life everywhere there is a random phenomenon. The random method is becoming more and more important in the processing of natural and social sciences mathematics model.Stochastic Volterra integral equations(SVIEs) have been widely used such as atmospheric and oceanic science, molecular biology, economics and so on; In many cases stochastic integral equations are difficult to obtain the analytic solution, then the numerical solution of SVIEs is particularly important.Firstly, the properties of the analytical solutions for the smooth SVIEs are analyzed,including the existence and uniqueness, the mean square boundedness, as well as analytical solution satisfying the Holder&& condition of the order of 1/2. The errors of the analytical solutions and the conditional expectations are estimated.We construct the collocation method for smooth SVIEs, and prove its solvability and boundedness. In order to analyze the convergence the method, we discuss the strong convergence order of the collocation solutions and the conditional expectations,using this results prove the strong convergence order of smooth SVIEs is investigated order, and numerical experiments are presented for verification.Finally, we consider the properties of the analytical solutions for non smooth SVIEs, and build collocation methods of non smooth SVIEs, and estimate drift error of the non smooth kernel SVIEs, then prove the strong convergence order is investigated.Examples are given to the corresponding verification.