Solving Problems of Optimal Stopping with Linear Costs of Observations

Abstract

In sequential analysis and starting with the seminal treatment of the optimality of the SPRT due to Wald and Wolfowitz [Wald, A.; Wolfowitz, J. Bayes solutions of sequential decision problems. Ann. Math. Statist. 1950, 21, 82–99] various problems of optimal stopping with linear costs of observation arise. Here, we shall describe a new method for solving such problems. For a payoff g(X t ) − ct we propose a linear representation of the form where M t is a local martingale, and the function h and the local martingale depend on a parameter λ. In the case of a diffusion process X t we shall show that, for a proper choice of λ, the boundary points of the optimal stopping region can be obtained from those points of the state space where the maxima of h are located. This method is inspired by a method of Beibel and Lerche [Beibel, M.; Lerche, H.R. A new look at optimal stopping problems related to mathematical finance. Statistica Sinica 1997, 7, 93–108; Beibel, M.; Lerche, H.R. Optimal stopping of regular diffusions under random discounting. Theory Probab. Appl. 2001, 45 (4), 547–557] who, for optimal stopping problems with discounted payoff, used a multiplicative representation e −rt g(X t ) = h(X t )M t with a suitable martingale.