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Evaluating first-passage probabilities for spectrally one-sided Lévy processes. (English) Zbl 0981.60048
The author’s aim is to compute numerically the first-passage-time distribution for a spectrally-negative Lévy process \(X\), which is characterized by \(E[\exp(zX_t)]= \exp(t\psi(z))\), where the Lévy exponent \(\psi\) has the representation \[ \psi(z)= {1\over 2} \sigma^2 z^2+ bz+ \int^0_{-\infty} \{e^{zx}- 1+ z(|x|\wedge 1)\} \nu(dx) \] and the measure \(\nu\) satisfies some integrability condition. To accomplish this, the author uses stable methods for inverting multidimensional Laplace transforms, which have been developed by J. Abate and W. Whitt [Queueing Syst. 10, No. 1/2, 5-87 (1992; Zbl 0749.60013) and ORSA J. Comput. 7, No. 1, 36-43 (1995; Zbl 0821.65085)].

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60G51 Processes with independent increments; Lévy processes
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