Research On Related Stopping Time Problems In Several Types Of Diffusion Processes

Diffusion processes are a very important content in stochastic processes,which are widely used in financial mathematics,economics,biology,electronic engineering and other fields.In the financial market,the trading time and pricing of various options are closely related to stopping time,therefore,studying the stopping time of diffusion processes has very important significance and application value for studying options and pricing.This paper mainly studies the problems of first hitting time,exit time and optimal stopping time in several diffusion processes,and the conclusions obtained are analyzed.In the chapter 1,the research background,research status of diffusion processes and the development and application of the stopping problems are mainly introduced.The research background includes several types of diffusion processes mainly studied in this paper: Vasicek model and diffusion risk model with drift coefficient.The stopping time problems involve first hitting time,exit time and optimal stopping time.Finally,the main research results and innovations of this paper are summarized.In the chapter 2,the first hitting time problem for the Vasicek model are mainly studied.Firstly,by using strong Markov property of the model,some explicit expression of the Laplace transform of the first hitting time are obtained.Secondly,these expressions are used to solve the probability and mathematical expectation of the first hitting time.Finally,numerical examples are given to illustrate the applications of the Laplace transform of the first hitting time.In the chapter 3,the exit time problem for the Vasicek model are mainly investigated.Firstly,by using strong Markov property of the model,some explicit expressions of the Laplace transform of the exit time are obtained.Secondly,these expressions are used to solve the probability and mathematical expectation of the exit time.Finally,different parameters are selected and the graph of the Laplace transform of exit time are shown.In the chapter 4,the optimal stopping problem of American put options for the diffusion risk model with drift coefficient are mainly considered.Firstly,by using the relevant conclusions of the first exit time of the model,the optimal stopping time formulation can be obtained.Secondly,the optimal value function and the optimal trading time are obtained.Finally,the graph of the value function under different parameters are presented.