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Some remarks on first passage of Lévy processes, the American put and pasting principles. (English) Zbl 1083.60034
Let \(X= (X_t)\) be a Lévy process defined on a filtered probability space \((\Omega,{\mathcal F},({\mathcal F}_t),P)\). For given constants \(K> 0\) and \(r\geq 0\) consider the following American put optimal stopping problem: \[ v(x)= \sup_\tau\,E_x[e^{-r\tau}(K- e^{X_\tau})^+]\tag{*} \] (the supremum taken over all \(({\mathcal F}_t)\)-stopping times \(\tau\)). Various authors [e.g. E. Mordecki, Finance Stoch. 6, No. 4, 473–493 (2002; Zbl 1035.60038)] have shown that for many Lévy processes (*) is solved by a strategy of the form \(\tau^*= \text{inf}\{t\geq 0: X_t< x^*\}\) for a specific value \(x^*<\log K\) so that \[ v(x)= KE_x[e^{-r\tau^*}]- E_x[e^{-r\tau^*+ X_{\tau^*}}]. \] The aim of the present paper is to comprehensively link a variety of identities for first passage problems of different Lévy processes and their connection to the American put optimal stopping problem and to explain precisely when smooth pasting is absent. In particular, it is thus possible to give alternative proofs of results obtained in Mordecki’s paper.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60J75 Jump processes (MSC2010)
91B70 Stochastic models in economics
60G51 Processes with independent increments; Lévy processes
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