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Optimal stopping problems in a stochastic and fuzzy system. (English) Zbl 0974.60024

Outgoing from a sequence of integrably bounded fuzzy random variables \((\widetilde{X}_n)_{n=0}^{\infty}\) which are defined in a usual manner by a measurability property of all of their \(\alpha\)-cuts, \(\alpha\in[0,1]\), a classical stopping time is defined with respect to the sequence \(({\mathcal M}_n)_{n=0}^{\infty}\) of \(\sigma\)-algebras, the smallest generated by the left and right ends \(\widetilde{X}_{k,\alpha}^-\) resp. \(\widetilde{X}_{k,\alpha}^+\) of their \(\alpha\)-cuts, \(k=0,\ldots,n\), \(\alpha\in[0,1]\). When handling with fuzzy random variables, as it is, one can also introduce fuzzy stopping times as \({\mathcal M}_n\)-measurable mappings \(\widetilde{\tau}:{\mathbb N}\times\Omega\rightarrow[0,1]\) (with some regularity conditions), the result of them gives the membership grade of stopping. For the known monotone case, an optimal fuzzy stopping time is explicitly given. For suitable payoff functions \(g:{\mathcal J}\rightarrow{\mathbb R}\), \({\mathcal J}\) being the set of all bounded closed subintervals of \({\mathbb R}\), and their evaluation \(E(G_{\tau})=\int_0^1g(E(\widetilde{X}_{\tau,\alpha}))d\alpha\), one can show that an optimal fuzzy stopping time \(\sigma^*\) majorizes a classical optimal stopping time \(\tau^*\), i.e. \(E(G_{\sigma^*})\geq E(G_{\tau^*})\). As an example, the fuzzification of the classical example of optimal stopping of an i.i.d. sequence with costs is discussed in great detail.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
26E50 Fuzzy real analysis
28E10 Fuzzy measure theory
11K45 Pseudo-random numbers; Monte Carlo methods
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References:

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