A class of strong convergence theorem for an arbitrarily adapted stochastic sequence without the non-decrease condition was established by using the truncation method of stochastic variables and the stop time.

By using the supplemantal variable method, the frequency transfer method and the renewal process theory, the Laplace Stieltjes Transform (LST) of the steady virtual waitingtime distribution of the model is obtained After getting the stationary distribution of the Queue Length, the average transient real waiting time of the model has also been obtained by using "stop time" and Wald s equation

Decision and Prediction of supply of goods in Markov theory are presented;optimization of supply of goods is also studied in stop time theory.

By using a stopping time,we discuss the fluctuations of stock price at the supper-critical state and sub-critical state.

This paper discusses a variation equation problem in a class of singular stochastic control with stopping time,gives its solution under two different conditions,which is a one order continuous differentiable and concave function,and gives the exact form.

The redeeming behavior can be described by stopping time,its possibility distribution and its expectation of present value of revenue are deduced.

In this paper, the concepts of the price of discrete-time America option and optimal stopping-time are defined, these concepts are discribed through moment-efficient, standard-market, combined investment, expectative-return and expectative price.

Based on the stopping-time theory and the martingale theory,an upper bound of the ruin probability is being proved.