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Notes on discrete compound Poisson model with applications to risk theory. (English) Zbl 1306.60050
Summary: Probability generating function (p.g.f.) is a powerful tool to study discrete compound Poisson (DCP) distribution. By applying inverse Fourier transform of p.g.f., it is convenient to numerically calculate probability density and do parameter estimation. As an application to finance and insurance, we firstly show that in the generalized CreditRisk\(^+\) model, the default loss of each debtor and the total default of all debtors are both approximately equal to a DCP distribution, and we give Le Cam’s error bound between the total default and a DCP distribution. Next, we consider geometric Brownian motion with DCP jumps and derive its \(r\)th moment. We establish the surplus process of the difference of two DCP distributions, and numerically compute the tail probability. Furthermore, we define the discrete pseudo compound Poisson (DPCP) distribution and give the characterizations and examples of DPCP distribution, including the strictly decreasing discrete distribution and the zero-inflated discrete distribution with \(P(X = 0) > 0.5\).

MSC:
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
91G40 Credit risk
62P05 Applications of statistics to actuarial sciences and financial mathematics
Software:
CreditRisk+
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