Generating Functions for Asymmetric Random Walk Processes With Double Absorbing Barriers

TL;DR

This paper derives generating functions for asymmetric random walks with two absorbing barriers, providing exact enumeration methods using advanced mathematical techniques like the Lagrange inversion formula.

Contribution

It introduces a novel application of the Lagrange inversion formula to derive generating functions for asymmetric walks with double barriers, enhancing analytical tools in stochastic process analysis.

Findings

Derived explicit generating functions for asymmetric walks with barriers

Applied Lagrange inversion to solve characteristic equations of the process

Provided exact enumeration formulas for constrained random walks

Abstract

Generating functions for asymmetric step-size paths restricted by two absorbing barriers are derived. The method begins by applying the Lagrange inversion formula to arbitrary powers of roots of the characteristic equation, that being a trinomial, which produces generating function as function (z) of the conditional probability of absorption of a particle, on a path restricted by two absorbing barriers. The exact enumeration of an asymmetric walk with two absorbing barriers is given.