Research On Stochastic Control Problems Driven By Geometric Brownian Motion

The first part of this paper studies mainly stochastic control models driven by geometric Brownian motion which 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 state of this model isHere, x>0,W₀ =0,μ,σare constant,μis drift,σis diffusion,σ≠0. The aim is to find a stopping timeÏ„^*∈T, which makesHere, T donates all the F_t stopping times. r > 0 , r is a discounted factor. (?)(x) is a function defined on (0,∞) which is an utility function . In this paper we put forward two kinds of utility functions, which areThe second part firstly gives a new proof of an important conclusion in stochastic control papers. Constructing function which satisfies some conditions is an important way to solve stochastic control problems. So this thought and method has reference value. Secondly, we give rigorous and detailed proof of some useful conclusions in some papers.