## The Annals of Probability

### Weak limits of perturbed random walks and the equation $Y_t = B_t + \alpha\sup\{Y_s\colon s \leq t\}+\beta\inf\{Y\sb s\colon s \leq t\}$

Burgess Davis

#### Abstract

Let $\alpha$ and $\beta$ be real numbers and $f \in C_0 [0, \infty)$. We study the existence and uniqueness of solutions g of the equation $g(t) = f(t) + \alpha \sup_{0 \leq s \leq t} g(s) + \beta \inf_{0 \leq s \leq t} g(s)$. Carmona, Petit, Le Gall, and Yor have shown existence or nonexistence and uniqueness for some $\alpha, \beta$. We settle the remaining cases. We study the nearest neighbor walk on the integers, which behaves just like fair random walk unless one neighbor has been visited and the other has not, when it jumps to the unvisited neighbor with probability p. If $p < 2/3$, we show these processes, scaled, converge to the solution of the equation above for Brownian paths, with $\alpha = \beta = (2p - 1)/p$.

#### Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 2007-2023.

Dates
First available in Project Euclid: 6 January 2003

https://projecteuclid.org/euclid.aop/1041903215

Digital Object Identifier
doi:10.1214/aop/1041903215

Mathematical Reviews number (MathSciNet)
MR1415238

Zentralblatt MATH identifier
0870.60076

#### Citation

Davis, Burgess. Weak limits of perturbed random walks and the equation $Y_t = B_t + \alpha\sup\{Y_s\colon s \leq t\}+\beta\inf\{Y\sb s\colon s \leq t\}$. Ann. Probab. 24 (1996), no. 4, 2007--2023. doi:10.1214/aop/1041903215. https://projecteuclid.org/euclid.aop/1041903215

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• WEST LAFAy ETTE, INDIANA 47907 E-MAIL: bdavis@snap.stat.purdue.edu