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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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