The Maximality Principle Revisited: On Certain Optimal Stopping Problems

We investigate in detail works of Peskir [15] and Meilijson [10] and develop a link between them. We consider the following optimal stopping problem: maximize V-tau = E[phi(S-tau) - integral(tau)(0) c(B-s)ds] over all stopping times with E integral(tau)(0)c(B-s)ds < infinity, where S = (S-t)(t >= 0) is the maximum process associated with real valued Brownian motion B, phi is an element of C-1 is non-decreasing and c >= 0 is continuous. From work of Peskir [15] we deduce that this problem has a unique solution if and only if the differential equation g'(s) = phi'(s)/2c(g(s))(s - g(s)) admits a maximal solution g(*)(s) such that g(*)(s) < s for all s >= 0. The stopping time which yields the highest payoff can be written as tau(*) = inf{t >= 0 : B-t <= g(*)(S-t)}. The problem is actually solved in a general case of a real-valued, time homogeneous diffusion X = (X-t : t >= 0) instead of B. We then proceed to solve the problem for more general functions phi and c. Explicit formulae for payoff are given. We apply the results to solve the so-called optimal Skorokhod embedding problem. We give also a sample of applications to various inequalities dealing with terminal value and maximum of a process.