On occupation times for a risk process with reserve-dependent premium

Abstract

Consider a risk reserve process under which the reserve can generate interest. For constants a and b such that a<b, we study the occupation time T _a,b (t), which is the total length of the time intervals up to time t during which the reserve is between a and b. We first present a general formula for piecewise deterministic Markov processes, which will be used for the computation of the Laplace transform of T _a,b (t). Explicit results are then given for the special case that claim sizes are exponentially distributed. The classical model is discussed in detail.

^*Research supported by RGC of Hong Kong SAR (Grant No. HKBU/2075/98P).

^**Research supported by NSF of China(Grant No. 19801020).

Keywords:

• Occupation time
• Piecewise deterministic Markov process
• Risk theory
• Duration of negative surplus
• Ruin

Acknowledgment

We thank the referee for helpful comments.

Notes

^*Research supported by RGC of Hong Kong SAR (Grant No. HKBU/2075/98P).

^**Research supported by NSF of China(Grant No. 19801020).