Optimal decision under ambiguity for diffusion processes

In this paper we consider stochastic optimization problems for an ambiguity averse decision maker who is uncertain about the parameters of the underlying process. In a first part we consider problems of optimal stopping under drift ambiguity for one-dimensional diffusion processes. Analogously to the case of ordinary optimal stopping problems for one-dimensional Brownian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in a two-parameter function space. In a second part we solve optimal stopping problems when the underlying process may crash down. These problems are reduced to one optimal stopping problem and one Dynkin game. Examples are discussed

Optimal stopping and hard terminal constraints applied to a missile guidance problem

This paper describes two new types of deterministic optimal stopping control problems: optimal stopping control with hard terminal constraints only and optimal stopping control with both minimum control effort And hard termind constraints. Both problems are initially formulated in continuous-time (a discretetime formulation is given towards the end of the paper) and soIutions given via dynamic programming. A numeric solution to the continuous-time dynamic programming equations is then briefly discussed. The optimal stopping with terminal constraints problem in continuous-time is a natural description of a particular type of missile guidance problem. This missile guidance appiication is introduced and the presented solutions used in missile engagements against targets

A class of recursive optimal stopping problems with applications to stock trading

In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. We then apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case.Comment: 35 pages, 2 figures. In this version, we provide a general analysis of a class of recursive optimal stopping problems with both finite-time and infinite-time horizon. We also discuss other application

Optimal stopping under probability distortion

We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general nonlinear function. The problem is inherently time inconsistent due to the Choquet integration involved. We develop a new approach, based on a reformulation of the problem where one optimally chooses the probability distribution or quantile function of the stopped state. An optimal stopping time can then be recovered from the obtained distribution/quantile function, either in a straightforward way for several important cases or in general via the Skorokhod embedding. This approach enables us to solve the problem in a fairly general manner with different shapes of the payoff and probability distortion functions. We also discuss economical interpretations of the results. In particular, we justify several liquidation strategies widely adopted in stock trading, including those of "buy and hold", "cut loss or take profit", "cut loss and let profit run" and "sell on a percentage of historical high".Comment: Published in at http://dx.doi.org/10.1214/11-AAP838 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org