Optimal Stopping of the Maximum Process : The Maximality Principle G

The solution is found to the optimal stopping problem with payoff sup E(S τ - ∫ 0 T c(X i )dt) where S = (S t ) t≥0 is the maximum process associated with the one-dimensional time-homogeneous diffusion X = (X t ) 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) = σ 2 (g(s))L'(g(s))/2c(g(s))(L(R)-L(g(s))) admits a maximal solution s → g * (s) which stays strictly below the diagonal in R 2 . [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 | X t < 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.