The Annals of Probability

Optimal stopping of the maximum process: the maximality principle

Goran Peskir

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Abstract

The solution is found to the optimal stopping problem with payoff $$\sup_{\tau} E(S_{\tau} - \int_0^{\tau} c(X_t)dt),$$ where $S = (S_t)_{t \geq 0}$ is the maximum process associated with the one-dimensional time-homogeneous diffusion $X = (X_t)_{t \geq 0}$, the function $x \mapsto c(x)$ is positive and continuous, and the supremum is taken over all stopping times $\tau$ of $X$ for which the integral has finite expectation. It is proved, under no extra conditions, that this problem has a solution; that is, the payoff is finite and there is an optimal stopping time, if and only if the following maximality principle holds: the first-order nonlinear differential equation $$g'(s) = \frac{\sigma^2 (g(s))L'(g(s))}{2c(g(s))(L(s) - L(g(s)))}$$ admits a maximal solution $s \mapsto g_*(s)$ which stays strictly below the diagonal in $\mathbb{R}^2$. [In this equation $x \mapsto \sigma(x)$ is the diffusion coefficient and $x \mapsto L(x)$ the scale function of $X$.] In this case the stopping time $$\tau_* = \inf{t > 0|X_t \leq g_*(S_t)}$$ is proved optimal, and explicit formulas for the payoff are given. The result has a large number of applications and may be viewed as the cornerstone in a general treatment of the maximum process.

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1614-1640.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855875

Digital Object Identifier
doi:10.1214/aop/1022855875

Mathematical Reviews number (MathSciNet)
MR1675047

Zentralblatt MATH identifier
0935.60025

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65] 34A34: Nonlinear equations and systems, general 35G15: Boundary value problems for linear higher-order equations 60E15: Inequalities; stochastic orderings
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G44: Martingales with continuous parameter 60J65: Brownian motion [See also 58J65] 34B05: Linear boundary value problems

Keywords
Optimal stopping maximum process diffusion process the principle of smooth fit the maximality principle optimal stopping boundary Doob and Hardy-Littlewood inequalities

Citation

Peskir, Goran. Optimal stopping of the maximum process: the maximality principle. Ann. Probab. 26 (1998), no. 4, 1614--1640. doi:10.1214/aop/1022855875. https://projecteuclid.org/euclid.aop/1022855875


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