The High Contact Principle as a Sufficiency Condition for Optimal Stopping

Abstract

The principle was first introduced by Samuelson (1965). He only gave a heuristic argument for the condition. In McKean's mathematical appendix to Samuelson's paper (McKean (1965)), a rigorous proof for the necessity of the condition was given for the case of a linear reward and a geometrical Brownian motion process, i.e. he proved that any solution of the optimal stopping problem has to satisfy the high contact condition. As there only exists one function satisfying the high contact condition in this situation, Samuelson's proposal is the only possible optimal solution. In Shiryayev (1978, Theorem 3.17) the condition appears as a necessary condition for one dimensional processes (but not under the name contact). Multidimensional versions of the theorem are given in Friedman (1976) and Bensoussan and Lions (1982). These theorems are derived from variational inequalities, and the assumptions in the theorems are too strong to apply to most economic applications. (See section 2.) The high contact principle is essentially a first order condition in the optimization of the stopping time (see Merton (1973, footnote 60), and for a rigorous further development of the same idea 0ksendal (1990)). To derive the second order conditions turns out to be easy, and we will in this paper prove, under weak conditions, that a solution proposal to an optimal stopping problem satisfying the high contact principle, is in fact an optimal solution to the problem. In this case we do not have to prove that there is only one solution satisfying the high contact principle, and the existence of the optimal solution is a part of the conclusion. Some results from stochastic analysis will be used without reference, these results can be found in 0ksendal (1989).

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