Optimal stopping of the maximum process: the maximality principle

The solution is found to the optimal stopping problem with payoff\τ E(S\τ - \∫₀\τ c(Xₜ)dt), where S = (Sₜ)t \≥0 is the maximum process associated with the one-dimensional time-homogeneous X = (Xₜ)t \≥ 0, the function x \↦ c(x) is positive and, and the supremum is taken over all stopping times \τ of for which the integral has finite expectation. It is proved, under no conditions, that this problem has a solution; that is, the payoff is and there is an optimal stopping time, if and only if the following principle holds: the first-order nonlinear differential g'(s) = \\σ² (g(s))L'(g(s))/2c(g(s))(L(s) - L(g(s))) a maximal solution s \↦ g_*(s) which stays strictly below the in ². [In this equation x \↦ \σ(x) is the coefficient and x \↦ L(x) the scale function of X.] In case the stopping time \τ_* = > 0|Xₜ \≤ g_*(Sₜ) is optimal, and explicit formulas for the payoff are given. The result has large number of applications and may be viewed as the cornerstone in a treatment of the maximum process.