An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space

Jeong-Gyoo Kim

doi:10.3390/math7111131

Abstract

We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process X t with one parameter t. If there is another varying key factor of an event described by a process, then we need to consider another parameter besides t. That is, we need a system of random variables with two parameters, say X s , t , which is far more complex than one-parameter processes. In this paper we challenge to develop such an idea, and provide the two-parameter version of an arc-sine law for a last hitting time. An arc-sine law for a two-parameter process is hardly found in literature. We use the properties of the two-parameter Wiener process for our development. Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed. One can use our results to predict an event happened in a system of random variables with two parameters, which is not available among existing arc-sine laws for one parameter processes.

Highlights

Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed
We develop a two-parameter version of an arc-sine law for a last hitting time, and this is achieved in the two-parameter Wiener space
We have developed a two-parameter process analogous to a last hitting time for the Wiener process and provided the probability distribution of the process

Summary

Introduction

We develop a two-parameter version of an arc-sine law for a last hitting time, and this is achieved in the two-parameter Wiener space. The existing arc-sine laws are about a stochastic process Xt with one parameter t, which is a time parameter in most applications. If there is another varying key factor of an event described. We devise a two-parameter process analogous to a last hitting time and develop the two parameter version of the arc-sine law using the properties of the two-parameter Wiener process. We briefly introduce the two-parameter Wiener space for background knowledge of our development.

Background

The Two-Parameter Version of an Arc-Sine Law for a Hitting Time

Conclusions