Present value distributions with applications to ruin theory and stochastic equations. (English) Zbl 0943.60098

Summary: We study the distribution of the stochastic integral \(\int ^{\infty }_0 e^{-R_t} dP_t\) where \(P\) and \(R\) are independent Lévy processes with a finite number of jumps on finite time intervals. The exact distribution is obtained in many special cases, and we derive asymptotic properties of the tails of the distributions in the general case. These results are applied to give two new examples of exact solutions of the probability of eventual ruin of an insurance portfolio where return on investments are stochastic. Finally we use the results to give new examples of exact solutions of the stochastic equations \(Z \overset \text{d} = AZ + B\) and \(Z \overset \text{d} = A(Z+C)\) where \(Z\) and \((A,B)\) (or \((A,C)\)) are independent.


60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Bateman, H., ()
[2] Chamayou, J.F.; Letac, G., Explicit stationary distributions for composition of random functions and products of random matrices, J. theoret. probab., 4, 3-36, (1991) · Zbl 0728.60012
[3] Dufresne, D., The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. actuarial J., 39-79, (1990) · Zbl 0743.62101
[4] Dufresne, D., On a property of gamma distributions, Bernoulli, 2, 287-291, (1996) · Zbl 0859.60064
[5] Feller, W., ()
[6] Fishman, G.S., Monte Carlo — concepts, algorithms, and applications, (1996), Springer New York · Zbl 0859.65001
[7] Goldie, C.M., Implicit renewal theory and tails of solutions of random equations, Ann. appl. probab., 1, 126-166, (1991) · Zbl 0724.60076
[8] Harrison, J.M., Ruin problems with compounding assets, Stochast. process. appl., 5, 67-79, (1977) · Zbl 0361.60053
[9] Jensen, J.L., Saddlepoint approximations, (1995), Oxford University Press Oxford · Zbl 1274.62008
[10] Lebedev, N.N., Special functions and their applications, (1972), Dover Publications New York · Zbl 0271.33001
[11] Liptser, R.S.; Shiryayev, A.N., Theory of martingales, (1989), Kluwer Dordrecht · Zbl 0728.60048
[12] Nilsen, T.; Paulsen, J., On the distribution of a randomly discounted compound Poisson process, Stochast. process. appl., 61, 305-310, (1996) · Zbl 0853.60048
[13] Norberg, R., Ruin problems with assets and liabilities of diffusion type, () · Zbl 0962.60075
[14] Paulsen, J., Risk theory in a stochastic economic environment, Stochast. process. appl., 46, 327-361, (1993) · Zbl 0777.62098
[15] Paulsen, J., Present value of some insurance portfolios, Scand. actuarial J., 11-37, (1997) · Zbl 0928.62103
[16] Sundt, B., An introduction to non-life insurance mathematics, (1993), Verlag Versicherungswirtschaft eV · Zbl 0811.62098
[17] Vervaat, W., On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Adv. appl. probab., 11, 750-783, (1979) · Zbl 0417.60073
[18] Yor, M., On some exponential functionals of Brownian motion, Adv. appl. probab., 24, 509-531, (1992) · Zbl 0765.60084
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