## Mahurin Honors College Capstone Experience/Thesis Projects

#### Department

Mathematics

#### Document Type

Thesis

#### Abstract

Within the four-sided boundaries of *x = 0*, *y = 0*, *x = m* and, *y = n*, a two-dimensional random walk begins at integer-valued coordinates *(h,k)* and moves one unit on each step either up, down, left, or right with non-zero probabilities that sum to 1. The process stops when hitting a boundary. Let *P(U), P(D), P(L)* and *P(R)* be the probabilities of hitting the upper, the lower, the left, and the right boundary first, respectively, when starting from a specific initial point within the boundaries. We use a Markov-Chain method to compute these probabilities. Let *x _{i,j}* be the probability of hitting the left boundary first, when starting at coordinates

*(i,j)*. We use a System of Equations method to find

*x*simultaneously for all possible starting points

_{i,j}*(i,j)*.

#### Advisor(s) or Committee Chair

Dr. David Neal

#### Disciplines

Mathematics

#### Recommended Citation

Wright, Miky, "Four-Sided Boundary Problem for Two-Dimensional Random Walks" (2013). *Mahurin Honors College Capstone Experience/Thesis Projects.* Paper 425.

https://digitalcommons.wku.edu/stu_hon_theses/425