Continuous-time Stopping Research Articles

General optimal stopping with linear cost

This article treats both discrete time and continuous time stopping problems for general Markov processes on the real line with general linear costs as they naturally arise in many problems in sequential decision making. Using an auxiliary function of maximum representation type, conditions are given to guarantee the optimal stopping time to be of threshold type. The optimal threshold is then characterized as the root of that function. For random walks, our results condense in the fact that all combinations of concave increasing payoff functions and convex cost functions lead to a one-sided solution. For Lévy processes, an explicit way to obtain the auxiliary function and the threshold is given by use of the ladder height processes. Lastly, the connection from discrete and continuous problems and possible approximation of the latter via the former is discussed and the results are applied to sequential tests of power one.

Nash Equilibria for Game Contingent Claims with Utility-Based Hedging

Game contingent claims (GCCs) generalize American contingent claims in allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we construct Nash equilibria by rewriting the game as a non-zero-sum stopping game in which players compare payoffs in terms of their exponential utility indifference values. As a by-product, we also obtain an existence result for the optimal exercise time of an American claim under utility indifference valuation by relating it to the corresponding nonlinear Snell envelope.

Construction of a Continuous Stopping Boundary from an Alpha Spending Function

Lan and DeMets (1983, Biometrika 70, 659-663) proposed a flexible method for monitoring accumulating data that does not require the number and times of analyses to be specified in advance yet maintains an overall Type I error, alpha. Their method amounts to discretizing a preselected continuous boundary by clumping the density of the boundary crossing time at discrete analysis times and calculating the resultant discrete-time boundary values. In this framework, the cumulative distribution function of the continuous-time stopping rule is used as an alpha spending function. A key assumption that underlies this method is that future analysis times are not chosen on the basis of the current value of the statistic. However, clinical trials may be monitored more frequently when they are close to crossing the boundary. In this situation, the corresponding continuous-time boundary should be used. Here we demonstrate how to construct a continuous stopping boundary from an alpha spending function. This capability is useful in the design of clinical trials. We use the Beta-Blocker Heart Attack Trial (BHAT) and AIDS Clinical Trials Group protocol 021 for illustration.