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August, 1982 Optimal Stopping on Autoregressive Schemes
Mark Finster
Ann. Probab. 10(3): 745-753 (August, 1982). DOI: 10.1214/aop/1176993782


For $\cdots \varepsilon_{-1}, \varepsilon_0, \varepsilon_1 \cdots$ i.i.d. random variables the autoregression $X_n = \varepsilon_n + a_1X_{n-1} + a_2X_{n-2} + \cdots$ yields a payoff $\gamma^n \sum^n_{-\infty} w_kX_k$ when stopped at time $n, 0 < \gamma < 1$ being the discount factor. The optimal rule is characterized and under certain restrictions is the first passage time $t = \inf \{n: X_n \geq c\}$. As $c \rightarrow \infty$ the distributions of $t$ and the remainder term $R_t = X_t - c$ are asymptotically independent and determined for exponential and algebraic tailed distributions on $\varepsilon_n$. An asymptotic expression for the optimal payoff is given and $c = c(\gamma)$ is calculated so that $t$ yields a payoff asymptotically optimal and asymptotic to $c$ as $\gamma \rightarrow 1$.


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Mark Finster. "Optimal Stopping on Autoregressive Schemes." Ann. Probab. 10 (3) 745 - 753, August, 1982.


Published: August, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0493.60050
MathSciNet: MR659543
Digital Object Identifier: 10.1214/aop/1176993782

Primary: 62L15
Secondary: 60G10 , 60G40 , 60J05 , 62P20

Keywords: asymptotic distribution , Autoregression , First passage time , optimal payoff

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • August, 1982
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