Evaluating first-passage probabilities for spectrally one-sided Lévy processes (Q2725318)

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 NEWLINE\[NEWLINE\psi(z)= {1\over 2} \sigma^2 z^2+ bz+ \int^0_{-\infty} \{e^{zx}- 1+ z(|x|\wedge 1)\} \nu(dx)NEWLINE\]NEWLINE 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 \textit{J. Abate} and \textit{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)].