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Optimal multiple stopping time problem. (English) Zbl 1235.60040
The authors show that computing the value function for the \(d\)-time optimal stopping problem \[ v(S)= \text{ess\,sup}\{\operatorname{E}[\psi(\tau_1, \tau_2,\dots, \tau_d)|{\mathcal F}_S],\,\tau_1,\dots, \tau_d\in T_S\} \] reduces to computing the value function for an optimal single stopping time problem \[ v(S)= \text{ess\,sup}\{\operatorname{E}[\phi(\theta)|{\mathcal F}_S],\,\theta\in T_S\} \] a.s. for a reward \(\phi\) expressible in terms of optimal \((d-1)\)-stopping time problems, i.e., by induction, the initial optimal \(d\)-stopping time problem can be reduced to nested optimal single stopping time problems. Sufficient regularity of \(\psi\) assumed, \(\phi\) is shown to have a progressively measurable càdlàg modification, which opens a way for interesting applications, e.g., in finance.

60G40 Stopping times; optimal stopping problems; gambling theory
60G07 General theory of stochastic processes
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
62L15 Optimal stopping in statistics
91G80 Financial applications of other theories
Full Text: DOI arXiv
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