## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 3 (1993), 763-776.

### On Ladder Height Distributions of General Risk Processes

Masakiyo Miyazawa and Volker Schmidt

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 3 (1993), 763-776.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005362

**Digital Object Identifier**

doi:10.1214/aoap/1177005362

**Mathematical Reviews number (MathSciNet)**

MR1233624

**Zentralblatt MATH identifier**

0780.60099

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Secondary: 60G10: Stationary processes 90B99: None of the above, but in this section

**Keywords**

Risk theory stationary marked point process severity of ruin ruin probability palm distribution inversion formula single-server queue

#### Citation

Miyazawa, Masakiyo; Schmidt, Volker. On Ladder Height Distributions of General Risk Processes. Ann. Appl. Probab. 3 (1993), no. 3, 763--776. doi:10.1214/aoap/1177005362. https://projecteuclid.org/euclid.aoap/1177005362