Cost of excursions until first crossing of the origin for random walks and Lévy flights: an exact general formula

We consider a discrete-time random walk on a line starting at \(x_0\geq 0\) where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary...

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Bibliographic Details
Published in:arXiv.org 2024-05
Main Authors: Mori, Francesco, Majumdar, Satya N, Vivo, Pierpaolo
Format: Article
Language:English
Subjects:
Cost analysis
Cost function
Mathematical models
Random walk
Online Access:Get full text
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Summary:We consider a discrete-time random walk on a line starting at \(x_0\geq 0\) where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy- and light-tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes, and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations.
ISSN:2331-8422