Optimal stopping for two-parameter processes

Abstract

A formalism for the optimal stopping of two-parameter processes is developed by analogy with the classical theory. An optimality criterion is established in terms of the conditional pay-off process. The problem of optimal stopping of a process indexed by IN2 is completely solved, in probabilistic terms, by using the notion of tactics. The method consis of searching for an optimal stopping point among the maximal stopping points up to which the Snell envelope is a martingale. On IR + 2 the difficulties arise from the lack of information about the behaviour of two-parameter supermartingales, and particularly the Snell envelope. The optimal stopping of a Brownian sheet is solved and we present the case of the bi-Brownian process. Associated systems of variational inequalities are proposed.