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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.

Suggested Citation

  • Ken Sennewald & Klaus Wälde, 2006. "“Itô’s Lemma“ and the Bellman Equation for Poisson Processes: An Applied View," CESifo Working Paper Series 1684, CESifo.
    • Handle: RePEc:ces:ceswps:_1684
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    Cited by:

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    5. 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.
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    7. 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.
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    9. 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.
    10. Lucas Bretschger & Alexandra Vinogradova, 2014. "Growth and Mitigation Policies with Uncertain Climate Damage," CESifo Working Paper Series 5085, CESifo.
    11. 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.
    12. Wälde, Klaus, 2011. "Production technologies in stochastic continuous time models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 616-622, April.
    13. 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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    More about this item

    Keywords

    stochastic differential equation; Poisson process; Bellman equation; portfolio optimization; consumption optimization;
    All these keywords.

    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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