The Economics Model Of The Geometric Brownian Motion With Poissonian Jumps

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 aren't thought to be a frequent but discrete jump process which are used to make models. In this process, most commonly, economic variables are seen as the combination of Brownian motion and Poissonian jump process. The dynamic process of economic variables is divided into continuous part and leaping part. Brownian motion is used to describe continuous part and the Poisson jumping process is used to describe the damage of the continuity which is caused by unpredictable random events.Three teachers Wang Zhiming,Huang Zhiyong and Xu Fangzhong from WuHan Science and Technology University issued a paper named《The Economics Model of the Geometric Brownian Motion with Poissonian Jumps》in《Mathematics Journal》in 2007, which discussed the economics model of the geometric Brownian motion with Poissonian jumps under the following conditionsin the case of the income function of R(x) = ax²-b,it obtains optimal solution of average income V+Sup E[e^-rtR(X₁)]by Ito formulas. Because the securitiesmovement has multiplicity, which just confined to the quadratic function is not in line with the actual situation, so further promotion is needed. On the basis of their paper, the power function R(x)=ax^a-b with the parameterαtakes place ofthe quadratic function and under the same limited conditions, optimal solution of average income is gained in order to construct a more general economic model. Actually, their conclusion is a special case of this essay, and the discussion in this essay is a generation of theirs and is more general.