On the existence of solutions of unbounded optimal stopping problems
type: Item
instance of: scholarly article
title: On the existence of solutions of unbounded optimal stopping problems
zbMATH Open document ID: 1337.60077
DOI: 10.1134/S0081543814080185
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2015-08-20T00:00:00Z
review text: The aim of the paper is to give sufficient conditions for the existence of solutions of optimal stopping problems for strong Markov processes when the payoff function is unbounded. This is a novelty: the previous literature has dealt only with problems where the payoff function was bounded in some sense. Moreover, the authors show that the solution can be represented as the first hitting time of the observable process. To be more concrete, consider the optimal stopping problem \[ V(x) \;= \;\sup_{\tau \in \mathcal{M}} \mathbb{E}_x G(X_\tau), \quad s \in S, \] where \(\mathcal{M}\) denotes the set of Markov times, \(S\) is a phase space and \(G:S \to \bar{\mathbb{R}}\) is the payoff function. The assumptions ``\(G\) is bounded from below'' and ``\(\mathbb{E}_x \sup_{t\geq 0} |G(X_t)| < \infty\) for any \(x \in S\)'' are replaced by a weaker condition and the optimal stopping time is given explicitly. Moreover, the authors generalize their main result to the case where the Markov process is not homogeneous or the payoff function \(G\) depends on the time by reducing the problem to one with a homogeneous process. Again, they give weak conditions under which there exists a solution. Lastly, the authors consider an optimal stopping problem for a Brownian motion and show by applying the previous results that there also exists a solution.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 6473893
zbMATH Keywords: optimal stopping problems
zbMATH Keywords: Markov processes
zbMATH Keywords: unbounded functions
zbMATH Keywords: Brownian motion
author: M. V. Zhitlukhin
author: Albert N. Shiryaev
MaRDI profile type: Publication
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cites work: Sequential Test for the Mean of a Normal Distribution III (Small $t$)
cites work: Sequential Tests for the Mean of a Normal Distribution IV (Discrete Case)
cites work: Q5641856
cites work: Q3658831
cites work: Q3378055
cites work: Q4663402
cites work: Q4172681
cites work: On Chernoff's hypotheses testing problem for the drift of a Brownian motion
full work available at URL: s0081543814080185
OpenAlex ID: W2080374567
description: scientific article; zbMATH DE number 6473893