Abstract
In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller--Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.
Highlights
Many practical optimal control problems could be expressed in terms of optimal stopping
As we show in this paper, the map w emerges naturally as a limit of finite horizon optimal stopping value functions
In Theorem 5.9 we show a condition for the uniqueness of a solution to the Bellman equation
Summary
Many practical optimal control problems could be expressed in terms of optimal stopping. The main contribution of this paper is the proof that both functions u and w are solutions to the associated optimal stopping Bellman equation. It can be shown that the Bellman equation admits a unique solution, which can be used to prove continuity of the function u ≡ w This result was one of the main building blocks used in [16], where the long-run impulse control problem was analysed. The main contribution of this part is Theorem 3.3, where we link the discrete time Bellman equation with the limits of suitable finite horizon stopping value functions. This is used, where we give a characterisation of solutions to the continuous time Bellman equation; see Theorem 5.2 for details.
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