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

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  • Ken Sennewald

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  • Klaus Wälde

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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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    References listed on IDEAS

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    1. Chang,Fwu-Ranq, 2009. "Stochastic Optimization in Continuous Time," Cambridge Books, Cambridge University Press, number 9780521541947, Fall.
    2. Aghion, Philippe & Howitt, Peter, 1992. "A Model of Growth through Creative Destruction," Econometrica, Econometric Society, vol. 60(2), pages 323-351, March.
    3. Obstfeld, Maurice, 1994. "Risk-Taking, Global Diversification, and Growth," American Economic Review, American Economic Association, vol. 84(5), pages 1310-1329, December.
    4. Tjalling C. Koopmans, 1963. "On the Concept of Optimal Economic Growth," Cowles Foundation Discussion Papers 163, Cowles Foundation for Research in Economics, Yale University.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Bassan, Bruno & Çinlar, Erhan & Scarsini, Marco, 1993. "Stochastic comparisons of Itô processes," Stochastic Processes and their Applications, Elsevier, vol. 45(1), pages 1-11, March.
    11. A. Sandmo, 1970. "The Effect of Uncertainty on Saving Decisions," Review of Economic Studies, Oxford University Press, vol. 37(3), pages 353-360.
    12. Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, vol. 87(1), pages 194-217, July.
    13. Moen, Espen R, 1997. "Competitive Search Equilibrium," Journal of Political Economy, University of Chicago Press, vol. 105(2), pages 385-411, April.
    14. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    15. 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.
    16. Duffie, Darrell & Fleming, Wendell & Soner, H. Mete & Zariphopoulou, Thaleia, 1997. "Hedging in incomplete markets with HARA utility," Journal of Economic Dynamics and Control, Elsevier, vol. 21(4-5), pages 753-782, May.
    17. Steger, Thomas M., 2005. "Stochastic growth under Wiener and Poisson uncertainty," Economics Letters, Elsevier, vol. 86(3), pages 311-316, March.
    18. 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.
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    Cited by:

    1. repec:eee:apmaco:v:276:y:2016:i:c:p:407-416 is not listed on IDEAS
    2. Levaggi, Rosella & Menoncin, Francesco, 2013. "Optimal dynamic tax evasion," Journal of Economic Dynamics and Control, Elsevier, vol. 37(11), pages 2157-2167.
    3. 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.
    4. 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.
    5. Lucas Bretschger & Alexandra Vinogradova, 2014. "Growth and Mitigation Policies with Uncertain Climate Damage," CESifo Working Paper Series 5085, CESifo Group Munich.
    6. Bauer, Christian, 2009. "The Reservation Wage under CARA and Limited Borrowing," Discussion Papers in Economics 10291, University of Munich, Department of Economics.
    7. repec:eee:wdevel:v:103:y:2018:i:c:p:323-335 is not listed on IDEAS
    8. 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.
    9. 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; C61; D81; D90; G11;

    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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