×

zbMATH — the first resource for mathematics

On the time to ruin and the deficit at ruin in a risk model with double-sided jumps. (English) Zbl 1153.91024
Summary: We consider a jump diffusion risk model, which consists of a Brownian motion, phase type distributed positive claims and general negative claims. The distributions of the time to ruin and the deficit at ruin will be studied by using Rouché’s Theorem, martingale and matrix analysis. We derive an explicit joint Laplace transform for the time to ruin and the deficit at ruin, as well as the Laplace transform for the time to ruin. Furthermore, our results still hold even when positive claims are rationally distributed.

MSC:
91B30 Risk theory, insurance (MSC2010)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Asmussen, S.; Avram, F.; Pistorius, M.R., Russian and American put option under exponential phase-type Lévy model, Stochastic process. appl., 109, 79-111, (2004) · Zbl 1075.60037
[2] Bladt, M.; Neuts, M.F., Matrix-exponential distributions: calculas and interpretations via flows, Stoch. models., 19, 1, 113-124, (2003) · Zbl 1020.60005
[3] Greene, R.E.; Krantz, S.G., Function theory of one complex variable, (1997), Wiley USA · Zbl 0997.32012
[4] Jacobsen, M., The time to ruin for a class of Markov additive risk process with two-sided jumps, Adv. appl. prob., 37, 963-992, (2005) · Zbl 1100.60021
[5] Kou, S.G.; Wang, H., First passage times of a jump diffusion process, Adv. appl. prob., 35, 504-531, (2003) · Zbl 1037.60073
[6] Perry, D.; Stadje, W.; Zacks, S., First-exit time for compound Poisson processes for some types of positive and negative jumps, Stoch. models., 18, 1, 139-157, (2002) · Zbl 0998.60089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.