Existence and explicit determination of optimal stopping times

We consider large classes of continuous time optimal stopping problems for which we establish the existence and form of the optimal stopping times. These optimal times are then used to find approximate optimal solutions for a class of discrete time problems.     

Risk-sensitive stopping problems for continuous-time Markov chains

In this paper we consider stopping problems for continuous-time Markov chains under a general risk-sensitive optimization criterion for problems with finite and infinite time horizon. More precisely our aim is to maximize the certainty equivalent of the stopping reward minus cost over the time horizon. We derive optimality equations for the value functions and prove the existence of optimal stopping times. The exponential utility is treated as a special case. In contrast to risk-neutral stopping problems it may be optimal to stop between jumps of the Markov chain. We briefly discuss the influence of the risk sensitivity on the optimal stopping time and consider a special house selling problem as an example.     

Optimal stopping problem in a model with compensated refusal of reward

We consider the optimal stopping problem with a possible compensated refusal of reward. We discuss functionals of exponential Brownian motion. The optimal stopping time is defined on the set of all finite stopping times. The functionals under consideration correspond to payments for standard American options.     

Representation results for jump processes with application to optimal stopping

It is shown that functions, measurable on the past of a jump process up to a stopping time, can be expressed as functions of the jump times and jump locations up to the stopping time. These results lead to formulas for conditional expectations with respect to the past of the process up to the stopping time. The use of these results is illustrated in giving a sufficient condition for optimality for optimal stopping of a partially observed jump Markov process.     

Dualité convexe,temps d'arrêt optimal et contrôle stochastique

The purpose of this paper is to apply convex analysis methods to prove existence results for optimal stopping time problems for the diffusion processes considered by Stroock and Varadhan. The work is based on the characterization given by Rost of the measures associated to the stopping times.The method is applied to the problem of control of diffusions where the stopping time is also a control.     

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.     

An entry-exit model involving construction and abandonment periods

We analyze, using the optimal stopping theory, the entry-exit decision on a project, which takes time to be constructed and abandoned. We obtain the closed-form expressions of optimal start time of entry, optimal start time of exit, and the maximal expected present value of the project. In addition, we examine the effects of construction and abandonment periods on the optimal start times of entry and exit.     

Optimal stopping for partially observed piecewise-deterministic Markov processes

This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an ?$?$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.