×

zbMATH — the first resource for mathematics

On nonlinear integral equations arising in problems of optimal stopping. (English) Zbl 1031.60030
Bakić, D. (ed.) et al., Functional analysis VII. Proceedings of the postgraduate school and conference, Dubrovnik, Croatia, September 17-26, 2001. Aarhus: University of Aarhus, Department of Mathematical Sciences, Var. Publ. Ser., Aarhus Univ. 46, 159-175 (2002).
Let \(\{B_t, 0\leq t\leq 1\}\) be a standard Brownian motion started at zero, let \(\lambda\geq 0\) be a given constant, and let \(G: [0,1]\times R^1\to R^1\) be a measurable function, satisfying regularity conditions. It is proved that the optimal stopping problem \[ V= \sup_{0\leq\tau\leq 1} E(e^{-\lambda\tau} G(\tau, B_\tau)), \] where \(\tau\) is a stopping time of \(B\), is solved by \(\tau_b= \inf\{0\leq t\leq 1\mid B_t\geq b_t\}\), where the optimal stopping boundary \(t\to b_t\) is characterized as a unique solution of the defined nonlinear integral equation. Some examples are discussed in detail.
For the entire collection see [Zbl 1004.00019].

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
35R35 Free boundary problems for PDEs
45G10 Other nonlinear integral equations
PDF BibTeX XML Cite