## The probability of ruin in finite time with discrete claim size distribution.(English)Zbl 0926.62103

Summary: The ruin time $$T$$ is considered as the time of first crossing between a compound Poisson trajectory and an upper increasing boundary. Under the assumption that the claim sizes are integer-valued, we show that the distribution of $$T$$ can be expressed in terms of generalized Appell polynomials. Using the algebraic properties of these polynomials elegant expressions are obtained for $$P(T>x)$$.

### MSC:

 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G35 Signal detection and filtering (aspects of stochastic processes)
Full Text:

### References:

 [1] Appell P. E., Ann. Sci. Ecole Norm. Sup. 9 pp 119– (1880) [2] Daniels H. E., Bull. Inter. Statist. Inst. 40 pp 994– (1963) [3] DOI: 10.2307/3212130 · Zbl 0232.60037 [4] DOI: 10.2307/3214516 · Zbl 0788.60068 [5] Gerber H. U., An introduction to mathematical risk theory (1979) · Zbl 0431.62066 [6] Kaz’min Y. A., Encyclopedia of math 1 pp 209– (1988) [7] Panjer M. H., Insurance risk models (1992) [8] DOI: 10.1016/0167-6687(94)00010-7 · Zbl 0806.62089 [9] Picard Ph., Adv. Appl. Prob. (1994) [10] Seal H., Stochastic theory of a risk business (1969) · Zbl 0196.23501 [11] DOI: 10.1215/S0012-7094-37-00347-8 · Zbl 0018.13601 [12] DOI: 10.1215/S0012-7094-39-00549-1 · Zbl 0022.01502 [13] DOI: 10.1007/BF02392231 · Zbl 0026.20805 [14] DOI: 10.1214/aoms/1177705056 · Zbl 0144.41603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.