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On risk reserve under distribution constraints

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EN
Abstrakty
EN
The purpose of this work is a study of the following insurance reserve model:
$R(t) = η + ∫_{0}^{t} p(s,R(s))ds + ∫_{0}^{t} σ(s,R(s))dW_{s} - Z(t)$, t ∈ [0,T],
P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0.
Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: $inf_{0≤t≤T} P{R(t) ≥ c} ≥ γ$ is considered.
Twórcy
  • Institute of Mathematics, Technical University, Podgórna 50, 65-246 Zielona Góra, Poland
Bibliografia
  • [1] J.P. Aubin and G. Da Prato, Stochastic viability and invariance, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), 595-613.
  • [2] J.P. Aubin and G. Da Prato, The viability theorem for stochastic differential inclusions, Stochastic Anal. Appl. 16 (1) (1998), 1-15.
  • [3] R. Bergmann and D. Stoyan, On exponential bounds for the waiting time distribution function in GI/G/1, Journal of Applied Probability 13 (1976), 411-417.
  • [4] P. Billingsley, Convergence of Probability Measures, J. Wiley and Sons, New York and London 1968.
  • [5] F. Dufresne and H.U. Gerber, Risk theory for the compound Poisson process that is perturbated by diffiusion, Insurance: Mathematics and Economics 10 (1991), 51-59.
  • [6] S. Gauthier and L. Thibault, Viability for constrained stochastic differential equations, Differential and Integral Equations 6 (6) 1395-1414.
  • [7] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer, Berlin-New York 1988.
  • [8] M. Kisielewicz, Viability theorems for stochastic inclusions, Discuss. Math. Diff. Incl. 15 (1) (1995), 61-75.
  • [9] T. Kurtz and Ph. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19 (3) (1991), 1035-1070.
  • [10] F. Lundberg, I, Approximerad Framställning av Sannonlikhetsfunktionen, II, A°terförsäkering av Kollektivresker, Almqvist and Wiksell, Upsala 1903.
  • [11] L. Mazliak, A note on weak viability for controlled diffusion, Prepublications du Lab. de Probab. Universite de Paris 6, 7, 513 (1999).
  • [12] M. Michta, A note on viability under distribution constrains, Discuss. Math. Algebra and Stochastic Methods 18 (2) (1998), 215-225.
  • [13] S.S. Petersen, Calculation of ruin probabilities when the premium depends on the current reserve, Scand. Act. J. (1990), 147-159.
  • [14] Ph. Protter, A connection between the expansion of filtrations and Girsanov`s theorem, Stochastic partial differential equations, Lecture Notes in Math. 1930 Springer, Berlin-New York, (1989), 221-224.
  • [15] Ph. Protter, Stochastic Integration and Differential Equations, Springer, Berlin-New York 1990.
  • [16] M. Yor, Grossissement de filtrations et absolue continuite de noyaux, Springer, Berlin-New York, Lecture Notes in Math. 1118 (1985), 6-15.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1015
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