On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. (English) Zbl 1049.60042

Pistorius, M. R.

J. Theor. Probab. 17, No. 1, 183-220 (2004).

Let \(X\) be a spectrally one-sided Lévy process and reflect it at its past infimum \(I\). That is the Lévy process minus its past infimum \(Y=X-I\). The author studies such a process. Here he determines the Laplace transform for \(Y\) if \(X\) is spectrally negative. Subsequently, he finds an expression for the resolvent measure for \(Y\) killed upon leaving \([0,a]\) and studies some properties of this process. Finally, the rate of convergence of the supremum of the reflected process to \(a\) is given.

Reviewer: Nicko G. Gamkrelidze (Moskva)

MSC:

60G51	Processes with independent increments; Lévy processes
60G70	Extreme value theory; extremal stochastic processes

Keywords:

reflected Lévy process; resolvent measure; Laplace transform