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The joint density function of three characteristics on jump-diffusion risk process. (English) Zbl 1066.91063
This paper is concerned with the study of the jump-diffusion risk process, i.e. the classical risk process that is perturbed by diffusion. More precisely, relying on a dissemination paper [H. U. Gerber, E. S. W. Shiu, ”The Joint Distribution of the Time of Ruin, the Surplus Immediately Before Ruin, and the Deficit at Ruin”. Insur. Math. Econ. 21, 129–137 (1997; Zbl 0894.90047)] the authors investigate and derive the explicit expression for the joint density function of the joint distribution of three important characteristics in the jump-diffusion risk process: the time of ruin, the surplus immediately before the ruin, and the deficit at ruin. The demonstration of the main result is based on a similar duality argument as in Gerber & Shiu (1997), and the strong Markov property to derive the explicit expression for an (intermediary) density function representing the probability that ruin does not occur before a certain time $$t$$, with the surplus $$x$$ belonging to the incremental interval at time $$t$$. Using this intermediary density function, some properties for Brownian motion, and basic ideas from the inspiring paper of Gerber & Shiu (1997), the authors succeed to obtain the explicit expression of the considered joint distribution. The resulted explicit formula for the joint density function of the three characteristics (viz. the time of ruin, the surplus immediately before the ruin, and the deficit at ruin) represents a generalization of the Dickson’s (1992) formula to the jump-diffusion risk model. Furthermore, there is also derived the distribution of the time that the negative surplus reaches the level zero firstly.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J75 Jump processes (MSC2010)
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##### References:
 [1] Dickson, D.C.M., On the distribution of the surplus prior to ruin, Insurance: mathematics and economics, 11, 191-207, (1992) · Zbl 0770.62090 [2] Dos Reis, A.D.E., How long is the surplus below zero?, Insurance: mathematics and economics, 12, 23-38, (1993) · Zbl 0777.62096 [3] Dufresne, F.; Gerber, H.U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: mathematics and economics, 10, 51-59, (1991) · Zbl 0723.62065 [4] Gerber, H.U.; Lantry, B., On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: mathematics and economics, 22, 263-276, (1998) · Zbl 0924.60075 [5] Gerber, H.U.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047 [6] Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, Astin bulletin, 17, 151-163, (1987) [7] Gihman, H.U., Skorohod, A.V., 1974. Theory of Stochastic Process II. Springer, New York. · Zbl 0291.60019 [8] Revuz, D., Yor, M., 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002 [9] Wang, G., A decomposition of ruin probability for the risk process perturbed by diffusion, Insurance: mathematics and economics, 28, 49-59, (2001) · Zbl 0993.60087 [10] Wu, R., Wang, G., Wei, L., 2003. Joint distributions of some actuarial random vectors containing the time of ruin. Insurance: Mathematics and Economics, submitted for publication. · Zbl 1024.62045 [11] Zhang, C.; Wu, R., Some results for the compound Poisson process that is perturbed by diffusion, Acta mathematicae sinica, English series, 18, 153-160, (2003) · Zbl 1004.60014
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