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Convergence of values in optimal stopping and convergence of optimal stopping times. (English) Zbl 1144.62067

In this paper the problem of stability of values of optimal stopping, and of optimal stopping times, under approximations of the process \(X\) are investigated [see D. Lamberton and G. Pagès, Sur l’approximation des réduites. Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 2, 331–355 (1990; Zbl 0704.60042)]. Let us consider a sequence \((X^n)_n\) of càdlàg processes which converges in probability to a càdlàg process \(X\). For all \(n\), let us denote by \({\mathcal F}^n\) the natural filtration of \(X^n\) and by \({\mathcal T}^n_T\) the set of \({\mathcal F}^n\) stopping times bounded by \(T\). Denote the values in the optimal stopping problem \(\Gamma^n(T)= \sup_{\tau\in {\mathcal T}^n_T}{\mathbf E}[\gamma(\tau,X_\tau^n)]\).
In this paper first are given conditions under which \(\Gamma^n(T)\) converges to \(\Gamma(T)\), and second, when it is possible to find a sequence \((\tau_n)\) of optimal stopping times w.r.t. the \(X^n\), to give further conditions under which the sequence \((\tau_n)\) converges to an optimal stopping time w.r.t \(X\). The results are obtained under hypothesis of inclusion of filtrations or convergence of filtrations.

MSC:

62L15 Optimal stopping in statistics
60G40 Stopping times; optimal stopping problems; gambling theory
60Fxx Limit theorems in probability theory
91A60 Probabilistic games; gambling
91A80 Applications of game theory

Citations:

Zbl 0704.60042
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