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October 1996 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
Ann. Probab. 24(4): 2007-2023 (October 1996). DOI: 10.1214/aop/1041903215

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 $\alpha = \beta = (2p - 1)/p$.

Citation

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Burgess Davis. "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 (4) 2007 - 2023, October 1996. https://doi.org/10.1214/aop/1041903215

Information

Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0870.60076
MathSciNet: MR1415238
Digital Object Identifier: 10.1214/aop/1041903215

Subjects:
Primary: 60F05, 60J15, 60J65, 82C41

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 4 • October 1996
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