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

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.

### 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.)
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### References:

 [1] DOI: 10.1016/0167-6687(91)90023-Q · Zbl 0723.62065 [2] DOI: 10.1016/0167-6687(94)00017-4 · Zbl 0814.62066 [3] DOI: 10.1214/aoap/1060202834 · Zbl 1046.62113 [4] Gerber HU, Schweiz. Verein. Versicherungsmath. Mitt. 69 pp 185– (1969) [5] Grandits P, Technical University (2003) [6] DOI: 10.1016/S0167-6687(00)00049-4 · Zbl 1007.91025 [7] DOI: 10.1007/s007800200095 · Zbl 1069.91051 [8] Højgaard B, University of Aalborg (2000) [9] Højgaard B Taksar M (1998) Optimal proportional reinsurance policies for diffusion models Scand. Actuarial J. 166 180 · Zbl 1075.91559 [10] DOI: 10.1016/S0167-6687(98)00007-9 · Zbl 1093.91518 [11] Palmowski Z, Bernoulli 8 pp 767– (2002) [12] DOI: 10.1002/9780470317044 [13] DOI: 10.1016/0167-6687(95)00003-B · Zbl 0837.62087 [14] Schmidli H, Scand. Actuarial J. pp 55– (2001) [15] Schmidli H, Working paper 176, Laboratory of Actuarial Mathematics, University of Copenhagen (2001) [16] DOI: 10.1214/aoap/1031863173 · Zbl 1021.60061 [17] Schmidli H, Manuscript, Laboratory of Actuarial Mathematics, University of Copenhagen (2002)
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