Numerical Solution Of Stochastic Differential Equations And Application Based On Stratonovich Pattern

People always use ordinary differential equations to describe and explain many phenomenaof the nature, the use of mathematical differential equation model solves many engineeringproblems by mathematical solution. Later, people consider that the random factors in somesystems can not be ignored, such as signal control system, financial system and biologicalsystem. Indeed, it is hardly to describe these phenomena without random item, then peoplerealize the importance of solutions of stochastic differential equations. However, analyticalsolution is not necessary in practice problem, and with the rapid development of computertechnology, numerical method can not need the theory and can be visually verified, it isconvenient to solve the solution of differential equation model. This paper researchesStratonovich stochastic differential equations which are a variant of It stochastic differentialequations, Its integral form is similar to traditional integral and It is not. This aroused theinterest of many people, people urgently want to know the nature and numerical method ofStratonovich stochastic differential equation. The main results of this thesis are as follows:1.In this paper, it uses numerical solution of It stochastic differential equations to reducecorresponding numerical solution of Stratonovich stochastic differential equations，such asEuler-Maruyama method, Back-Euler method, Trapezoidal method and Misltein method.Then it proves convergence of numerical solution and gives its stability region.2.This paper constructs the numerical solution by direct Taylor expansion. In order to avoidcomplex higher derivative and integral, it constructs four stage explicit Runge-kutta methodby using color tree theory. The convergence order of the method is1, it is a new method andhas high precision.3.In the stock market, many interest rates models, bond price models and derivativesecurities models conform to stochastic differential equations. It uses the Stratonovich typestochastic differential equations to simulate these practical problems, and then use thenumerical solution of Stratonovich stochastic differential equations to determine the interestrate and bond prices, it is significant to do theoretical study for the actual. In order to solvecommon problems of interest rate model, by using the corresponding numerical method andcombining the two fork tree theory，this paper proposes a fast estimation rate algorithm， which can determine the stock price exactlly.