Appl Math Optim 50:1–20 (2004)
2004 Springer-Verlag New York, LLC
An Analogue of the Cram´er–Lundberg Approximation in the
Optimal Investment Case
Institut f¨ur Finanz- und Versicherungsmathematik, TU Wien,
Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
Communicated by I. Karatzas
Abstract. We consider ruin probabilities for an insurance company, which can
also invest in the stock market. The risk process is modeled by a compound Poisson
process and the stock price by geometric Brownian motion. We show that if the
tails of the claims are light tailed, then the optimal strategy is asymptotically given
by holding a constant $-value in the stock position. Furthermore, we show that a
kind of Cram´er–Lundberg approximation holds for the minimal ruin probability.
Everything is shown under assumptions, which are analogous to the assumptions in
the case of the classical Cram´er–Lundberg approximation without investment.
Key Words. Optimal investment, Ruin probabilities, Integro-differential equa-
AMS Classiﬁcation. Primary 45J05, 91B30, Secondary 26A12.
We start with a brief description of the classical Cram´er–Lundberg model. Consider an
insurance company whose surplus at time t is described by R(t), and set R(0) = s.As
the wealth in the future is unknown, we describe it by a stochastic process. The ﬁrst
ingredient stems from the incoming premiums. We assume that the premium intensity
This research was supported by the “Austrian Science Foundation” (Fonds zur F¨orderung der wis-
senschaftlichen Forschung), Project No. P15603.