Optimal double stopping time problem (Q847103)

The optimal double stopping problem is investigated (see \textit{G. W.~Haggstrom} [Ann. Math. Stat. 38, 1618--1626 (1967; Zbl 0189.18301)] for the classical formulation of the problem for the double indexed discrete time processes and \textit{R.~Ehjdukyavichyus} [Litov. Mat. Sb. 22, No. 3, 209--215 (1982; Zbl 0515.60045)] for continuous time Markov processes). Let \((\Omega,{\mathcal F},({\mathcal F}_t)_{0\leq t\leq T}\) be the probability space with filtration. Assuming that for every stopping times \(\tau\), \(\sigma\) with respect to the filtration given the double stopping process \(\psi(\tau,\sigma)\) is the \({\mathcal F}_{\tau\vee \sigma}\) measurable, positive, random variable [see \textit{N.~El Karoui}, Les aspects probabilistes du contrôle stochastique. Ecole d'ete de probabilites de Saint-Flour IX-1979, Lect. Notes Math. 876, 74--238 (1981; Zbl 0472.60002)]. The aim is to determine \(v(S) = \text{ess\; sup}_{\tau _1 ,\tau _2 \geq S}E [\psi (\tau _1 ,\tau _2)|\mathcal F_s]\) for each stopping time \(S\). Following the optimal one stopping time problem it is proved that the optimal stopping times exist and a method is given how to compute them. The key point is the construction of a new reward \(\phi\) such that \(v(s) = \text{ess sup}\{E[\phi (\tau )|\mathcal F_s],\tau \geq S\}\).