Optimal stopping of the maximum process: The maximality principle

Abstract

The solution is found to the optimal stopping problem with payoff supτE(Sτ - ∫0τ c(Xt)dt), where S = (St)t≥0 is the maximum process associated with the one-dimensional time-homogeneous diffusion X = (Xt)t≥0, the function x → c(x) is positive and continuous, and the supremum is taken over all stopping times τ 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) = σ2(g(s))L′(g(s))/2c(g(s))(L(s)-L(g(s))) admits a maximal solution s → g*(s) which stays strictly below the diagonal in ℝ2. [In this equation x → σ(x) is the diffusion coefficient and x → L(x) the scale function of X.] In this case the stopping time τ* = inf{t>0|Xt≤g*(St)} 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.