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On approximative solutions of optimal stopping problems. (English) Zbl 1235.60038
In a series of papers (see, e.g., R. Kühne and L. Rüschendorf [Theory Probab. Appl. 48, No. 3, 465–480 (2003) and Teor. Veroyatn. Primen. 48, No. 3, 557–575 (2003; Zbl 1079.60043)]) an approximation method was developed in order to solve approximatively optimal stopping problem for sequences \(X_1,\dots, X_n\) of independent or dependent random variables by some limiting stopping problems for Poisson and related point processes. The basic assumption in this approach is convergence of the imbedded planar point process \[ N_n= \sum^n_{i=1} \delta_{({i\over n}, X^n_i)}@>D>> N\tag{\(*\)} \] to some Poisson (or related) point process \(N\). Here, \(X^n_i= (X_i- b_n)/a_n\) is a suitable normalization of \(X_i\). For the process \(N\) in \((*)\) which has accumulation points along a lower boundary curve, an optimal stopping problem in continuous time can be formulated. The optimal solution for the latter is given by a threshold stopping time, the threshold function being determined by a first-orer differential equation. In Section 2 it is shown that (under certain assumptions) the optimal stopping curve \(u\) solves a differential equation of the form \[ u'(t)=- \int^\infty_{u(t)} G(t,y)\,dy,\quad 0\leq t< 1,\;u(1)= c\tag{\(**\)} \] for some guarantee value \(-\infty\leq c<\infty\). Here, \(G\) (called the “intensity function”) is defined explicitly via the intensity measure of \(N\). If \(c\) is finite, \((**)\) has a unique solution. If \(c=-\infty\), the authors show that the optimal stopping curve is the maximal solution of \((**)\). In Section 3 it is shown that \((**)\) can be solved in “explicit form” if \(G\) satisfies a certain “separation condition”. In Section 4 an approximation result by R. Kühne and L. Rüschendorf (loc. cit.) is extended to dependent sequences. In Section 5 optimal stopping of sequences of the form \[ X_i= c_i Z_i+ d+i \] is discussed in a fairly complete form. Here, J\((Z_i)\) is an i.i.d. sequence, and the discount factors \((c_i)\) and the observation costs \((d_i)\) fulfill some criteria to ensure point process convergence.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
62L15 Optimal stopping in statistics
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