Optimal Stopping on Autoregressive Schemes

Abstract

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 $\gamma \rightarrow 1$.