## The Annals of Probability

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

### Optimal stopping of the maximum process: the maximality principle

#### 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