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On the distribution of a randomly discounted compound Poisson process

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  • Nilsen, Trygve
  • Paulsen, Jostein

Abstract

We study the distribution of the stochastic integral [integral operator]0t8e-Rt dPt where R is a Brownian motion with positive drift and P is an independent compound Poisson process. We show that in the special case when the jumps of P are exponentially distributed, the integral has the same distribution as that of a gamma variable divided by an independent beta variable.

Suggested Citation

  • Nilsen, Trygve & Paulsen, Jostein, 1996. "On the distribution of a randomly discounted compound Poisson process," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 305-310, February.
    • Handle: RePEc:eee:spapps:v:61:y:1996:i:2:p:305-310
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    References listed on IDEAS

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    1. Harrison, J. Michael, 1977. "Ruin problems with compounding assets," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 67-79, February.
    2. Paulsen, Jostein, 1993. "Risk theory in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 327-361, June.
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    Cited by:

    1. Gjessing, Håkon K. & Paulsen, Jostein, 1997. "Present value distributions with applications to ruin theory and stochastic equations," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 123-144, October.
    2. Kardaras, Constantinos & Robertson, Scott, 2017. "Continuous-time perpetuities and time reversal of diffusions," LSE Research Online Documents on Economics 67495, London School of Economics and Political Science, LSE Library.
    3. Anita Behme & Alexander Lindner, 2015. "On Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 681-720, June.
    4. Hao, Xuemiao & Tang, Qihe, 2008. "A uniform asymptotic estimate for discounted aggregate claims with subexponential tails," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 116-120, August.
    5. Bae, Taehan & Kim, Changki & Kulperger, Reginald J., 2009. "Securitization of motor insurance loss rate risks," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 48-58, February.
    6. Zhang, Zhehao, 2019. "On the stochastic equation L(Z)=L[V(X+Z)] and properties of Mittag–Leffler distributions," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 365-376.
    7. Klüppelberg, Claudia & Kostadinova, Radostina, 2008. "Integrated insurance risk models with exponential Lévy investment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 560-577, April.
    8. Constantinos Kardaras & Scott Robertson, 2017. "Continuous-time perpetuities and time reversal of diffusions," Finance and Stochastics, Springer, vol. 21(1), pages 65-110, January.

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