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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.

##### MSC:
 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
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