Asymptotics of ruin probabilities for controlled risk processes in the small claims case. (English) Zbl 1087.62116

Hipp, Christian; Schmidli, Hanspeter

Scand. Actuar. J. 2004, No. 5, 321-335 (2004).

The authors consider an extension of the classical Cramér-Lundberg risk model which admits the possibility for insurers to invest some amount into a risky asset, which is modeled by a geometric Brownian motion. The main task of the paper is to investigate the asymptotic behaviour (as initial capital \(u\to\infty\) ) of the ruin probability \(\psi(u)\) under the optimal investment strategy in the small claims case. For the mentioned model the Lundberg inequalities and the Cramér-Lundberg approximation for \(\psi(u)\) are proved. This is done by combining properties of \(\psi(u)\) derived from the relevant Hamilton-Jacobi-Bellman equation, a martingale approach and change of measure method. In addition, convergence of the optimal investment strategy is studied. In particular, it is proved that the optimal strategy converges as \(u\to\infty\) to the value that optimizes the adjustment coefficient.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

62P05	Applications of statistics to actuarial sciences and financial mathematics
91B30	Risk theory, insurance (MSC2010)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30	Applications of stochastic analysis (to PDEs, etc.)
93E20	Optimal stochastic control
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

ruin probability; change of measure; optimal control; Cramér-Lundberg approximation; adjustment coefficient; Lundberg bounds; martingale; geometric Brownian motion

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