Optimal Stopping Variables for Brownian Motion

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.