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First passage time for compound Poisson processes with diffusion: ruin theoretical and financial applications. (English) Zbl 1401.91160

Summary: In this paper, we propose to revisit Kendall’s identity (see, e.g. D. G. Kendall [J. R. Stat. Soc., Ser. B 19, 207–212 (1957; Zbl 0118.35502)]) related to the distribution of the first passage time for spectrally negative Lévy processes. We provide an alternative proof to Kendall’s identity for a given class of spectrally negative Lévy processes, namely compound Poisson processes with diffusion, through the application of Lagrange’s expansion theorem. This alternative proof naturally leads to an extension of this well-known identity by further examining the distribution of the number of jumps before the first passage time. In the process, we generalize some results of H. U. Gerber [Insur. Math. Econ. 9, No. 2–3, 115–119 (1990; Zbl 0731.62153)] to the class of compound Poisson processes perturbed by diffusion. We show that this main result is particularly relevant to further our understanding of some problems of interest in actuarial science. Among others, we propose to examine the finite-time ruin probability of a dual Poisson risk model with diffusion or equally the distribution of a busy period in a specific fluid flow model. In a second example, we make use of this result to price barrier options issued on an insurer’s stock price.

MSC:

91B30 Risk theory, insurance (MSC2010)
91G20 Derivative securities (option pricing, hedging, etc.)
60J75 Jump processes (MSC2010)
60G51 Processes with independent increments; Lévy processes
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