An Optimal Stopping Time Research On The Finite-step Random Walk

Random walk is one of the simplest and the most important random processes in special case. In the random processes and most existing literature, the random walks are involved in infinite-steps and based on the Law of Large Numbers. However, in many real applications, especialy in the fields of economics, biology and gambling games, we have to deal with the finite-step random walk with absorbent boundaries.Optimal Stopping Problem (OSP) of random walks were found to have a considerable number of important applications in real fields such as networks, stock market, stochastic control, especially in the field of gambling game, which has attracted more and more researchers.This paper proposes a gambling game of finite-step simple random walk with absorbent boundaries. We address a problem of optimal stop, which is defined as the absorbent boundary value with maximum profit.We make this problem more simple with the help of the mathematics model on the axis. Based on the classical probability computation, we give out the solving process of this problem when the value of M is small, then we discuss the expression of the profit function and the optimal stopping time K* when N>M or N≤M, at the same time,we give out the simulations with the help of computer, which makes the problem more visual and specific. Finally, we discuss the relationship of random walk and the Brownian motion which give out a train of thought to the small sample behavior of the Brownian motion.