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"Ito's Lemma" and the Bellman equation for Poisson processes: An applied view

Author

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  • Sennewald, Ken
  • Wälde, Klaus
Abstract
Rare and randomly occurring events are important features of the economic world. In continuous time they can easily be modeled by Poisson processes. Analyzing optimal behavior in such a setup requires the appropriate version of the change of variables formula and the Hamilton-Jacobi-Bellman equation. This paper provides examples for the application of both tools in economic modeling. It accompanies the proofs in Sennewald (2005), who shows, under milder conditions than before, that the Hamilton-Jacobi-Bellman equation is both a necessary and sufficient criterion for optimality. The main example here consists of a consumption-investment problem with labor income. It is shown how the Hamilton-Jacobi-Bellman equation can be used to derive both a Keynes-Ramsey rule and a closed form solution. We also provide a new result.

Suggested Citation

  • Sennewald, Ken & Wälde, Klaus, 2005. ""Ito's Lemma" and the Bellman equation for Poisson processes: An applied view," W.E.P. - Würzburg Economic Papers 58, University of Würzburg, Department of Economics.
    • Handle: RePEc:zbw:wuewep:58
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    References listed on IDEAS

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    6. 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.
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    9. 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.
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    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.
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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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