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A Risk Model with an Observer in a Markov Environment

Author

Listed:
  • Hansjörg Albrecher

    () (Department of Actuarial Science, University of Lausanne, Lausanne CH-1015, Switzerland
    Swiss Finance Institute, University of Lausanne, Lausanne CH-1015, Switzerland)

  • Jevgenijs Ivanovs

    () (Department of Actuarial Science, University of Lausanne, Lausanne CH-1015, Switzerland)

Abstract

We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration.

Suggested Citation

  • Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, Open Access Journal, vol. 1(3), pages 1-14, November.
  • Handle: RePEc:gam:jrisks:v:1:y:2013:i:3:p:148-161:d:30342
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    References listed on IDEAS

    as
    1. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    2. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.
    3. Mandjes, Michel & Ivanovs, Jevgenijs & Kella, Offer & D'Auria, Bernardo, 2010. "First passage of a Markov additive process and generalized Jordan chains," DES - Working Papers. Statistics and Econometrics. WS ws103923, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Ivanovs, Jevgenijs, 2013. "A note on killing with applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 29-34.
    5. Albrecher, Hansjörg & Lautscham, Volkmar, 2013. "From Ruin to Bankruptcy for Compound Poisson Surplus Processes," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 43(02), pages 213-243, May.
    6. Gerber, Hans U. & Lin, X. Sheldon & Yang, Hailiang, 2006. "A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 36(02), pages 489-503, November.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Jin, Can & Li, Shuanming & Wu, Xueyuan, 2016. "On the occupation times in a delayed Sparre Andersen risk model with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 304-316.
    2. repec:eee:spapps:v:127:y:2017:i:8:p:2699-2724 is not listed on IDEAS
    3. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    4. repec:eee:spapps:v:128:y:2018:i:1:p:255-290 is not listed on IDEAS
    5. Choi, Michael C.H. & Cheung, Eric C.K., 2014. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 121-132.
    6. Feng, Runhuan & Shimizu, Yasutaka, 2014. "Potential measures for spectrally negative Markov additive processes with applications in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 11-26.
    7. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.

    More about this item

    Keywords

    Markov additive process; level-crossing probabilities; Poissonian observation; ruin probability; occupation times;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law

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