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On the stochastic equation L(Z)=L[V(X+Z)] and properties of Mittag–Leffler distributions

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  • Zhang, Zhehao

Abstract

The stochastic equation Z=dV(X+Z), where V, X and Z are independent, has a wide range of applications in finance, insurance, telecommunications and time series analysis. Dufresne[8,9] solves for some specific cases of this equation by the algebraic properties of beta and gamma distributions. This paper aims to generalise Dufresne’s results to beta and Mittag–Leffler distributions and solve for new specific distributions of Z.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:365-376
    DOI: 10.1016/j.amc.2019.05.003
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    References listed on IDEAS

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    1. 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.
    2. Chamayou, J.-F.Jean-François, 2004. "Products of double gamma, gamma and beta distributions," Statistics & Probability Letters, Elsevier, vol. 68(2), pages 199-208, June.
    3. Zhang, Zhehao, 2019. "New Results On The Distribution Of Discounted Compound Poisson Sums," ASTIN Bulletin, Cambridge University Press, vol. 49(1), pages 169-187, January.
    4. Leveille, Ghislain & Garrido, Jose, 2001. "Moments of compound renewal sums with discounted claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(2), pages 217-231, April.
    5. R. Pillai, 1990. "On Mittag-Leffler functions and related distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 157-161, March.
    6. 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.
    7. Paulsen, Jostein, 1993. "Risk theory in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 327-361, June.
    8. Zhang, Zhehao, 2018. "Renewal sums under mixtures of exponentials," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 281-301.
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