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August, 1993 On Ladder Height Distributions of General Risk Processes
Masakiyo Miyazawa, Volker Schmidt
Ann. Appl. Probab. 3(3): 763-776 (August, 1993). DOI: 10.1214/aoap/1177005362

Abstract

We consider a continuous-time risk process $\{Y_a(t); t \geq 0\}$ defined for a stationary marked point process $\{(T_n,X_n)\}$, where $Y_a(0) = a$ and $Y_a(t)$ increases linearly with a rate $c$ and has a downward jump at time $T_n$ with jump size $X_n$ for $n \in \{1,2,\ldots\}$. For $a = 0$, we prove that, under a balance condition, the descending ladder height distribution of $\{Y_0(t)\}$ has the same form as the case where $\{(T_n,X_n)\}$ is a compound Poisson process. This generalizes the recent result of Frenz and Schmidt, in which the independence of $\{T_n\}$ and $\{X_n\}$ is assumed. In our proof, a differential equation is derived concerning the deficit $Z_a$ at the ruin time of the risk process $\{Y_a(t)\}$ for an arbitrary $a \geq 0$. It is shown that this differential equation is also useful for proving a continuity property of ladder height distributions.

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Masakiyo Miyazawa. Volker Schmidt. "On Ladder Height Distributions of General Risk Processes." Ann. Appl. Probab. 3 (3) 763 - 776, August, 1993. https://doi.org/10.1214/aoap/1177005362

Information

Published: August, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0780.60099
MathSciNet: MR1233624
Digital Object Identifier: 10.1214/aoap/1177005362

Subjects:
Primary: 60K30
Secondary: 60G10 , 90B99

Keywords: inversion formula , Palm distribution , Risk theory , ruin probability , severity of ruin , single-server queue , stationary marked point process

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.3 • No. 3 • August, 1993
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