Numerical Method For Stochastic Capital System

In recent years, the capital system has attracted a wide attention. With the influence of random factors such as technical progress, natural disasters in real questions, the stochastic capital system can perfectly reflect the nature's phenomenon. However, in most cases, it is difficult to get the analytic solutions, therefore, the numerical method is a powerful tool for the model and its properties.This paper discusses the stochastic capital system and the main contents are shown as following:The first part, constructing a numerical scheme of the model by using Euler method,then combining with the existence and uniqueness theorems of generalized Khasminskiitype, we investigate the numerical solutions of stochastic differential delay capital system convergence to the real solutions in mean sense and in probability. Finally, an example is given to verify the algorithm of this paper.The second part, by applying a class of backward Euler methods for stochastic agedependent capital system, we obtain the asymptotic mean-square boundedness of numerical solutions with the conditions of that the drift coefficient and the diffusion coefficient are one-sided Lipschitz and bounded.The third part, the Brownian motion and Poisson process to the capital system is introduced and we construct the numerical solution with Taylor approximation method, finally, by using Burkholder-Davis-Gundy inequality, H¨lder inequality and Doob martingale inequality, The convergence of the numerical solutions is proved.