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June, 1983 Optimal Stopping in the Stock Market When the Future is Discounted
Mark Finster
Ann. Statist. 11(2): 564-568 (June, 1983). DOI: 10.1214/aos/1176346161


In the random walk stock market model, a stock is purchased at price $x$ and is sold at time $t$ for the price $x + S_t$ where $S_t = \sum^t_0 X_i, X_i$ is the price change during the $i$th epoch, and $X_1, X_2, \cdots$ are i.i.d. random variables with $\mu = E(X_1) > 0$ and finite $\sigma^2 = E(X^2_1) - \mu^2 > 0$. Discounting the future by a factor of $\gamma$ per epoch, $0 < \gamma < 1$, a selling or stopping policy $t$ has expected payoff or utility $u(t) = E\{\gamma^t(x + S_t)\}$. This article determines second order asymptotic properties of the optimal selling policy $s$, the first passage time of $S_n$ across a straight line boundary $c$, whose utility is equal to the value $V(x) = \sup_tu(t)$ of the stock purchased at price $x$. Specifically, as $\gamma \rightarrow 1$, renewal theory is utilized to evaluate the limiting distribution of $s, E(s), V(x)$, and the first passage boundary $c$ up to second order terms.


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Mark Finster. "Optimal Stopping in the Stock Market When the Future is Discounted." Ann. Statist. 11 (2) 564 - 568, June, 1983.


Published: June, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0529.60042
Digital Object Identifier: 10.1214/aos/1176346161

Primary: 62L15
Secondary: 60G40 , 60J15 , 60K05

Keywords: excess over the boundary , first passage times , Random walk , renewal theory , sequential analysis

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 2 • June, 1983
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