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.