A Study On The Optimal Stopping Problem For Simple Diusion Processes

The optimal stopping problem is extensively applied in the economy, finance andstatisticses realm. by the concave function concept gave a new characteristics of the ex-cessive function for one dimensioal di?usion processes. Under some conditions,this newcharacterization shown that excessivity is equivalent to concavity.this permits a characterization of the value function of the optimal stopping problem as"the smallest non-negative concave majorant"of the reward function.Dayanik and Karatzas studied optimal stopping problem for one dimensioal di?usionprocesses and get a some main result. we based on their theory and to generalize theirresults, namely, when left boundary of the di?usion processes X in the state space I isnatural, we make use of Ito? theorem ,local martingale,strong Markov property , lowersemi- continuous function and Fatou's lemma to certificateβ-excessive function and F-concavity of the value function, discussed the su?cient and necessary condition of thevalue function's finity ,and continuty of value function . thus, gave the calculation methodof the value function and optimal stopping time.at the end,the results are illustratedin Karatzas and Ocone's optimal stopping problem and Guo and Shepp's Risk-AverseInvestors option,also calculated the value function and optimal regions.