Random walks maximizing the probability to visit an interval

TL;DR

This paper investigates the optimal strategies for random walks with bounded differences to maximize the probability of visiting a specified interval, providing explicit solutions and extending to super-martingales.

Contribution

It offers a solution to maximize the visit probability for bounded difference martingales and extends the results to super-martingales.

Findings

Identifies the optimal random walk strategies for maximizing visit probability.

Provides explicit formulas for the maximum visit probability.

Extends the analysis to super-martingale processes.

Abstract

We consider random walks, say $W_n=(M_0, M_1,\dots, M_n)$, of length $n$ starting at 0 and based on the martingale sequence $M_k$ with differences $X_m=M_m-M_{m-1}$. Assuming that the differences are bounded, $|X_m|\leq 1$, we solve the problem \begin{equation} D_n(x)\=\sup P \left\{W_n \ \text{visits an interval}\ [x,\infty)\right\},\qquad x\in R, \label{piirma} \end{equation} where $\sup$ is taken over all possible $W_n$. In particular, we describe random walks which maximize the probability in $\eqref{piirma}$. We also extend the result to super-martingales.