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The use of vector-valued martingales in risk theory

  • Original Research Paper
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Blätter der DGVFM

Abstract

A completely dependent risk process with perturbation and phase-type distributed claim sizes is analyzed. Claim arrivals are modeled by a Markovian arrival process. Using a vector-valued martingale, the Laplace transform of the time to ruin is derived algorithmically. The conditional memoryless property of the phase-type distribution yields the distribution of the deficit at ruin as a corollary.

Zusammenfassung

Wir untersuchen einen perturbierten Risikoprozess mit stochastisch abhängigen Schadenszeiten und -forderungen. Erstere werden durch einen Markovschen Ankunftsprozess (MAP) modelliert, die Höhe der letzteren durch Verteilungen vom Phasentyp. Mittels eines vektorwertigen Martingals wird die Laplace-Transformierte der Ruinzeit bestimmt. Die bedingte Gedächtnislosigkeit der Phasentypverteilung ermöglicht dann auch die Bestimmung der Defizitverteilung zum Ruinzeitpunkt.

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References

  1. Gerber HU (1970) An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuarietidskrift 53:205–210

    Google Scholar 

  2. Dufresne F, Gerber HU (1991) Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance: Math Econ 10:51–59

    Article  MATH  MathSciNet  Google Scholar 

  3. Schmidli H (1995) Cramer–Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Insurance: Math Econ 16:135–149

    Article  MATH  MathSciNet  Google Scholar 

  4. Gerber HU, Landry B (1998) On a discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Math Econ 22:263–276

    Article  MATH  MathSciNet  Google Scholar 

  5. Wang G (2001) A decomposition of the ruin probability for the risk process perturbed by diffusion. Insurance: Math Econ 28:49–59

    Article  MATH  Google Scholar 

  6. Tsai CCL (2003) On the expectations of the present values of the time of ruin perturbed by diffusion. Insurance: Math Econ 32:413–429

    Article  MATH  Google Scholar 

  7. Chiu SN, Yin CC (2003) The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion. Insurance: Math Econ 33(1):59–66

    Article  MATH  MathSciNet  Google Scholar 

  8. Li S (2006) The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion. Scand Actuarial J 2:73–85

    Article  Google Scholar 

  9. Li S, Garrido J (2005) The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion. Scand Actuarial J 3:161–186

    Article  MathSciNet  Google Scholar 

  10. Gerber HU, Shiu ESW (1998) On the time value of ruin. North Am Actuarial J 2(1):48–78

    MATH  MathSciNet  Google Scholar 

  11. Albrecher H, Boxma OJ (2005) On the discounted penalty function in a Markov-dependent risk model. Insurance: Math Econ 37(3):650–672

    Article  MATH  MathSciNet  Google Scholar 

  12. Lu Y, Li S (2005) On the probability of ruin in a Markov-modulated risk model. Insurance: Math Econ 37:522–532

    Article  MATH  MathSciNet  Google Scholar 

  13. Lu Y (2006) On the severity of ruin in a Markov-modulated risk model. Scand Actuarial J 4:183–202

    Google Scholar 

  14. Asmussen S (2000) Ruin Probabilities. Singapore: World Scientific

  15. Badescu AL, Breuer L, Da Silva Soares A, Latouche G, Remiche M-A, Stanford D (2005a) Risk processes analyzed as fluid queues. Scand Actuarial J 2:127–141

    Article  MathSciNet  Google Scholar 

  16. Badescu AL, Breuer L, Drekic S, Latouche G, Stanford D (2005b) The surplus prior to ruin and the deficit at ruin for a correlated risk process. Scand Actuarial J 6:433–445

    Article  MathSciNet  Google Scholar 

  17. Ramaswami V (2006) Passage times in fluid models with application to risk processes, Methodol Computing Appl Probab 8(4):497–515

    Google Scholar 

  18. Ahn S, Badescu AL (2007) On the analysis of the Gerber–Shiu discounted penalty function for risk processes with Markovian arrivals. Insurance: Math Econ 41(2):234–249

    Article  MATH  MathSciNet  Google Scholar 

  19. Neuts MF (1979) A versatile Markovian point process. J Appl Probab 16:764–779

    Article  MATH  MathSciNet  Google Scholar 

  20. Asmussen S (1995) Fluid flow models with or without Brownian motion. Stoch Models 11:1–20

    Article  MathSciNet  Google Scholar 

  21. Asmussen S, Kella O (2000) Multi-dimensional matringales. Adv Appl Prob 32:376–393

    Article  MATH  MathSciNet  Google Scholar 

  22. Gershgorin S (1931) Über die Abgrenzung der Eigenwerte einer Matrix. Izv Akad Nauk USSR Otd Fiz-Mat Nauk 7:749–754

    Google Scholar 

  23. Latouche G, Ramaswami V (1999) Introduction to Matrix Analytic Methods in Stochastic Modeling. Philadelphia: ASA SIAM

  24. De Smit JHA (1983) The queue GI/M/s with customers of different types or the queue GI/Hm/s. Adv Appl Probab 15:392–419

    Article  MATH  Google Scholar 

  25. De Smit JHA (1995) Explicit Wiener–Hopf factorization for the analysis of multi-dimensional queues. Advances in Queueing: Theory, Methods and Open Problems, JH Dshalalow, CRC Press, Boca Raton

  26. Gaver D (1966) Observing stochastic processes, and approximate transform inversion. Operations Res 14(3):444–459

    Article  MathSciNet  Google Scholar 

  27. Stehfest H (1970) Numerical inversion of Laplace transforms. Commun ACM 13(1):45–47

    Article  Google Scholar 

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Authors and Affiliations

  1. Toronto, Canada

    Andrei Badescu

  2. Kent, UK

    Lothar Breuer

Authors
  1. Andrei Badescu
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  2. Lothar Breuer
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Correspondence to Andrei Badescu.

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Badescu, A., Breuer, L. The use of vector-valued martingales in risk theory . Blätter DGVFM 29, 1–12 (2008). https://doi.org/10.1007/s11857-008-0049-z

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  • DOI: https://doi.org/10.1007/s11857-008-0049-z

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