The Financial Model Of Geometric Brown Motion With Jump And It’s Properties

Brown movement as a continuous time parameter and the continuous state space of a random process, is one of the most basic, the most simple and most importantly, random process. It is yet to know most clearly, one of the most colorful random process properties.In economics, the Geometric Brown motion can represent project value, output, price, input costs and imbalances at random with time,and actively influence the dynamic change process of the investment decision variables. There is plenty of evidence that the Geometric Brown motion model can’t get all the characteristics of the evolution of the Stock price. One of the evidence is the Stock price in the unpredictable time suddenly "jump". Generally, Brown as everywhere continuous diffusion process, the economic variables as infrequent but the process to establish the model of discrete jump, and one of the most common is the economic variables as a mixture of Brown motion and Poisson jump, its dynamic process can be divided into infrequent part and "jump", with Brown motion to describe the continuous part, at the same time, unpredictable random events on the continuity of the damage are described by Poisson jump. Study motivation of this paper is to build a profit function for logarithmic function with jump Financial model of Geometric Brown motion,and obtain optimal yield conditions.This article of Financial Model of Geometric Brown Motion with Poisson jump: N=(Nt; t∈R+) is a Poisson process with the parameters λ＞0, T=(Ti;i∈Z+) is a time sequence when Poisson process Nt occurs.In the case of the revenue function R(x)=ln(x),we obtained the optimal average earningsV=SupE［e-rtR(Xt)］with Ito formula.