The Numerical Solution Of Stochastic Differential Equations

The models of stochastic differential equations have been widely used in engineering,finance,biology and other disciplines,but the exact solutions of stochastic differential equations are not often available.At this time,we can only find the approximate numerical solutions to replace the exact solutions.Therefore,the numerical solution of the stochastic differential equation is brought to the attention of the scholars.However,now most of the numerical solution method can effectively solve the low dimensional stochastic differential equation numerical solution,the approximate equation dimension when it rises,the complexity of the existing numerical method will increase exponentially,calculate the approximate numerical solution accuracy will not be able to guarantee.In this paper,a deep learning neural network algorithm is introduced to obtain numerical solutions of high-dimensional stochastic differential equations,and black-scholes equation of 50 dimensions is taken as an example to establish a four-layer neural network using Tensor Flow framework to obtain approximate numerical solutions.In the first chapter,the origin and development of stochastic differential equations are introduced.In the second chapter,we introduces the Brownian motion,mainly introduces the definition and properties of Brownian motion,and simulates the trajectory of onedimensional standard Brownian motion with R software.Then we introduce it? formula and stochastic differential equation,and prove the existence and uniqueness of the solution of stochastic differential equation.In the third chapter,we introduces two common numerical methods for stochastic differential equations: Euler-Maruyama method and Milstein method,and introduces the convergence and stability of the two numerical methods.Then R software is used to simulate the approximate numerical solution of one dimensional stochastic differential equation.In the fourth chapter,we introduces the relationship between kolmogorov's partial differential equation and stochastic differential equation,and then introduces an algorithm to calculate the numerical solution of high-dimensional stochastic differential equation with deep learning neural network.