The Annals of Probability

Optimal Stopping Variables for Brownian Motion

LeRoy H. Walker

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Abstract

For a constant $\beta > \frac{1}{2}$ and $W$ normalized Brownian motion with parameter space the nonnegative real line, the stopping variable $\lambda$ defined by $$\lambda = \sup \{t: W(s) < y_0(1 + s)^{\frac{1}{2}}, 0 \leqq s < t\}$$ where $y_0$ is the unique positive root of $$\int^\infty_0 x^{2(\beta-1)}e^{(yx-x^2/2)} dx = y \int^\infty_0 x^{(2\beta-1)}e^{(yx-x^2/2)} dx$$ is shown to be optimal in the sense that $E\{(1 + \lambda)^{-\beta}W(\lambda)\}$ is equal to the supremum of $E\{(1 + \tau)^{-\beta}W(\tau)\}$ over all stopping variables $\tau$ with respect to $W$. The values of $y_0$ for $\beta =$ 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 are given.

Article information

Source
Ann. Probab., Volume 2, Number 2 (1974), 317-320.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996711

Digital Object Identifier
doi:10.1214/aop/1176996711

Mathematical Reviews number (MathSciNet)
MR397867

Zentralblatt MATH identifier
0282.60029

JSTOR
links.jstor.org

Subjects
Primary: 62L15: Optimal stopping [See also 60G40, 91A60]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J65: Brownian motion [See also 58J65]

Keywords
Optimal stopping variables (rules) normalized Brownian motion

Citation

Walker, LeRoy H. Optimal Stopping Variables for Brownian Motion. Ann. Probab. 2 (1974), no. 2, 317--320. doi:10.1214/aop/1176996711. https://projecteuclid.org/euclid.aop/1176996711


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