A model of finite-step random walk with absorbent boundaries

Abstract

This paper proposes a model of finite-step lattice random walk with absorbent boundaries. We address a problem of optimal stop for this model, which is defined as the absorbent boundary value with maximum profit. Compared with many existing optimal stop investigations in the random process, our study only considers the small-sample behaviour (i.e., small number of steps behaviour) and does not consider the limit behaviour of the walk. The optimal stop time is given based on classical probability computation. Since the small-sample is more practical and common than the large-sample in many real world problems, the result obtained in this paper may provide some useful guidelines for real applications associated with the finite-step random walk such as the stock market and gambling games.

Keywords:

• classical probability model
• small sample
• lattice random walk
• random walk with finite-step restriction
• optimal stop of random walk

2000 AMS Subject Classification:

• 82B41

Acknowledgements

This research is supported by the National Natural Science Foundation of China (60473045). Hebei Hi-tech project (04213533) and the plan of 100 excellent innovative scientists of the first group in the Education Department of Hebei Province.