The Study Of The Economic Model Of The Geometric Brownian Motion With Poissonian Jumps

Brownian motion which is considered as a stochastic process of continuous state space and time parameters is a basic,simple and the most important stochastic processes. Many other random processes are often considered as its functional or promotion of some significance. It is so far the most clearly understanding stochastic process the nature of which is most colorful. Brownian motion has been widely popular in many purely scientific fields, such as physics,economics,communications theory,biology,management science and mathematical statistics. Meanwhile, because Brownian motion is closely associated with differential equations, it has become an important channel to contact probability with analysis.In economics, geometric Brownian motion can be said as a dynamic change process that the project value,output prices,and input costs with the passage of time initiatively and randomly affect the variables of investment decisions. Generally, brown process is considered as the diffusion process for everywhere continuum, but the reality is that the economic variables which is thought the not frequent but discrete jump process are used to make model, in this process most commonly economic variables as the mixed of Brownian motion and Poisson jumping process. The dynamic process of economic variables is separated to continuous part and leap part, Brown is used to describe continuous movement, and the Poisson jump process is used to describe the damage which unpredictable random events often do with this continuity.This paper discusses the economic model of the geometric Brownian motion withIn the case of income function of R ( x)=ax~2b and R ( x)= ax~2-b, it obtains optimal solution of average income using the Ito equation.