Research On Stochastic Control Problems Driven By Geometric Brownian Motion

This paper studies mainly stochastic control models driven by geometric Brownian motion. The first model is optimal stopping problem. The objective is to find a stopping time for maximizing the expected benefit. In order to solve this problem, we rely on stochastic calculus method via variational equation. Under a suitable set of sufficient conditions, we prove the existence of an optimal stopping time.The second model is impulse control problem. Our objective is to find the strategy of an admissible impulse control when no intervention. Since we want to minimize the expected total discounted cost function, We should consider only those strategies for which cost function is well defined and finite. First, let us consider the strategy of no intervention. Hence, by relying on stochastic calculus method, we get the strategy of no intervention is admissible.The state is geometric Brownian motion in two models of this paper. The geometric Brownian motion has many practical application in the study of economics and finance. It is very important to study the stochastic control problems driven by geometric Brownian motion in practice.