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“Itô's Lemma” and the Bellman Equation for Poisson Processes: An Applied View

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  • Ken Sennewald

  • Klaus Wälde

Abstract

Using the Hamilton-Jacobi-Bellman equation, we derive both a Keynes-Ramsey rule and a closed form solution for an optimal consumption-investment problem with labor income. The utility function is unbounded and uncertainty stems from a Poisson process. Our results can be derived because of the proofs presented in the accompanying paper by Sennewald (2006). Additional examples are given which highlight the correct use of the Hamilton-Jacobi-Bellman equation and the change-of-variables formula (sometimes referred to as “Ito’s-Lemma”) under Poisson uncertainty.
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Suggested Citation

  • Ken Sennewald & Klaus Wälde, 2006. "“Itô's Lemma” and the Bellman Equation for Poisson Processes: An Applied View," Journal of Economics, Springer, vol. 89(1), pages 1-36, October.
    • Handle: RePEc:kap:jeczfn:v:89:y:2006:i:1:p:1-36
    DOI: 10.1007/s00712-006-0203-9
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    Cited by:

    1. Brausmann, Alexandra & Bretschger, Lucas, 2018. "Economic development on a finite planet with stochastic soil degradation," European Economic Review, Elsevier, vol. 108(C), pages 1-19.
    2. Song, Bo & Zhang, Ya & Park, Ju H. & Huang, Huan, 2016. "L2–L∞ filtering for stochastic systems driven by Poisson processes and Wiener processes," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 407-416.
    3. Levaggi, Rosella & Menoncin, Francesco, 2013. "Optimal dynamic tax evasion," Journal of Economic Dynamics and Control, Elsevier, vol. 37(11), pages 2157-2167.
    4. Fissel, Benjamin E & Glibert, Ben, 2010. "Exogenous Productivity Shocks and Capital Investment in Common-pool Resources," University of California at San Diego, Economics Working Paper Series qt1qp1g9ts, Department of Economics, UC San Diego.
    5. Augeraud-Véron, Emmanuelle & Fabbri, Giorgio & Schubert, Katheline, 2021. "Prevention and mitigation of epidemics: Biodiversity conservation and confinement policies," Journal of Mathematical Economics, Elsevier, vol. 93(C).
    6. Juan Carlos Parra-Alvarez & Olaf Posch & Mu-Chun Wang, 2017. "Estimation of Heterogeneous Agent Models: A Likelihood Approach," CESifo Working Paper Series 6717, CESifo.
    7. Bauer, Christian, 2011. "On the reservation wage under CARA and limited borrowing," Mathematical Social Sciences, Elsevier, vol. 62(2), pages 126-129, September.
    8. Lucas Bretschger & Alexandra Vinogradova, 2014. "Growth and Mitigation Policies with Uncertain Climate Damage," CEEES Paper Series CE3S-02/14, European University at St. Petersburg, Department of Economics.
    9. Tsuboi, Mizuki, 2019. "Resource scarcity, technological progress, and stochastic growth," Economic Modelling, Elsevier, vol. 81(C), pages 73-88.
    10. Thomas Douenne, 2020. "Disaster Risks, Disaster Strikes, and Economic Growth: the Role of Preferences," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 38, pages 251-272, October.
    11. Cieślik, Andrzej & Goczek, Łukasz, 2018. "Control of corruption, international investment, and economic growth – Evidence from panel data," World Development, Elsevier, vol. 103(C), pages 323-335.
    12. Georg Müller-Fürstenberger & Ingmar Schumacher, 2009. "Uncertainty and Insurance in Endogenous Climate Change," Research Papers in Economics 2009-02, University of Trier, Department of Economics.
    13. Wälde, Klaus, 2011. "Production technologies in stochastic continuous time models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 616-622, April.
    14. Juan Carlos Parra-Alvarez & Olaf Posch & Mu-Chun Wang, 2017. "Identification and estimation of heterogeneous agent models: A likelihood approach," CREATES Research Papers 2017-35, Department of Economics and Business Economics, Aarhus University.

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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