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Abstract
We study the optimal stopping time problem $v(S)={\rm ess}\sup_{\theta \geq S} E[\phi(\theta)|\mathcal{F}_S]$, for any stopping time $S$, where the reward is given by a family $(\phi(\theta),\theta\in\mathcal{T}_0)$ \emph{of non negative random variables} indexed by stopping times. We solve the problem under weak assumptions in terms of integrability and regularity of the reward family. More precisely, we only suppose $v(0) < + \infty$ and $(\phi(\theta),\theta\in \mathcal{T}_0)$ upper semicontinuous along stopping times in expectation. We show the existence of an optimal stopping time and obtain a characterization of the minimal and the maximal optimal stopping times. We also provide some local properties of the value function family. All the results are written in terms of families of random variables and are proven by only using classical results of the Probability Theory
Highlights
In the present work we study the optimal stopping problem in the setup of families of random variables indexed by stopping times, which is more general than the classical setup of processes
An optimal stopping problem can be naturally expressed in terms of families of random variables indexed by stopping times
The family (φ(θ), θ ∈ T0) of random variables indexed by stopping times is called the reward family
Summary
In the present work we study the optimal stopping problem in the setup of families of random variables indexed by stopping times, which is more general than the classical setup of processes. In the present work, which is self-contained, we study the general case of a reward given by an admissible family φ = (φ(θ), θ ∈ T0) of non negative random variables, and we solve the associated optimal stopping time problem only in terms of admissible families. Using this approach, we avoid the aggregation step as well as the use of Mertens’ decomposition. We use the following notation: for real valued random variables X and Xn, n ∈ N, “Xn ↑ X” stands for “the sequence (Xn) is nondecreasing and converges to X a.s.”
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