IDEAS home Printed from https://ideas.repec.org/p/zbw/wuewep/58.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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, Chair for Monetary Policy and International Economics.
  • Handle: RePEc:zbw:wuewep:58
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/22352/1/wep58.pdf
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    as
    1. Aghion, Philippe & Howitt, Peter, 1992. "A Model of Growth through Creative Destruction," Econometrica, Econometric Society, vol. 60(2), pages 323-351, March.
    2. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    3. Gene M. Grossman & Elhanan Helpman, 1991. "Quality Ladders in the Theory of Growth," Review of Economic Studies, Oxford University Press, vol. 58(1), pages 43-61.
    4. Sennewald, Ken, 2005. "Controlled Stochastic Differential Equations under Poisson Uncertainty and with Unbounded Utility," Dresden Discussion Paper Series in Economics 03/05, Technische Universität Dresden, Faculty of Business and Economics, Department of Economics.
    5. Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, vol. 87(1), pages 194-217, July.
    6. Kiyotaki, Nobuhiro & Wright, Randall, 1991. "A contribution to the pure theory of money," Journal of Economic Theory, Elsevier, vol. 53(2), pages 215-235, April.
    7. Moen, Espen R, 1997. "Competitive Search Equilibrium," Journal of Political Economy, University of Chicago Press, vol. 105(2), pages 385-411, April.
    8. Framstad, Nils Chr. & Oksendal, Bernt & Sulem, Agnes, 2001. "Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 233-257, April.
    9. Walde, Klaus, 1999. "A Model of Creative Destruction with Undiversifiable Risk and Optimising Households," Economic Journal, Royal Economic Society, vol. 109(454), pages 156-171, March.
    10. Steger, Thomas M., 2005. "Stochastic growth under Wiener and Poisson uncertainty," Economics Letters, Elsevier, vol. 86(3), pages 311-316, March.
    11. Aase, Knut Kristian, 1984. "Optimum portfolio diversification in a general continuous-time model," Stochastic Processes and their Applications, Elsevier, vol. 18(1), pages 81-98, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Levaggi, Rosella & Menoncin, Francesco, 2013. "Optimal dynamic tax evasion," Journal of Economic Dynamics and Control, Elsevier, vol. 37(11), pages 2157-2167.
    2. 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.
    3. Juan Carlos Parra-Alvarez & Olaf Posch & Mu-Chun Wang, 1710. "Identification and estimation of heterogeneous agent models: A likelihood approach," CREATES Research Papers 2017-35, Department of Economics and Business Economics, Aarhus University.
    4. Lucas Bretschger & Alexandra Vinogradova, 2014. "Growth and Mitigation Policies with Uncertain Climate Damage," CESifo Working Paper Series 5085, CESifo Group Munich.
    5. Bauer, Christian, 2009. "The Reservation Wage under CARA and Limited Borrowing," Discussion Papers in Economics 10291, University of Munich, Department of Economics.
    6. 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.
    7. Wälde, Klaus, 2011. "Production technologies in stochastic continuous time models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 616-622, April.

    More about this item

    Keywords

    Stochastic differential equation; Poisson process; Bellman equation; Portfolio optimization; Consumption optimization;

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zbw:wuewep:58. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (ZBW - German National Library of Economics). General contact details of provider: http://edirc.repec.org/data/wfwuede.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.