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

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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. --

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Bibliographic Info

Paper provided by University of Würzburg, Chair for Monetary Policy and International Economics in its series W.E.P. - Würzburg Economic Papers with number 58.

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Date of creation: 2005
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Handle: RePEc:zbw:wuewep:58

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

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  1. 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.
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  7. Nobuhiro Kiyotaki & Randall Wright, 1989. "A contribution to the pure theory of money," Staff Report, Federal Reserve Bank of Minneapolis 123, Federal Reserve Bank of Minneapolis.
  8. Duffie, Darrell & Fleming, Wendell & Soner, H. Mete & Zariphopoulou, Thaleia, 1997. "Hedging in incomplete markets with HARA utility," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 21(4-5), pages 753-782, May.
  9. Aghion, P. & Howitt, P., 1989. "A Model Of Growth Through Creative Destruction," Working papers, Massachusetts Institute of Technology (MIT), Department of Economics 527, Massachusetts Institute of Technology (MIT), Department of Economics.
  10. Bassan, Bruno & Çinlar, Erhan & Scarsini, Marco, 1993. "Stochastic comparisons of Itô processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 45(1), pages 1-11, March.
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  12. Sandmo, Agnar, 1970. "The Effect of Uncertainty on Saving Decisions," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 37(3), pages 353-60, July.
  13. Sennewald, Ken, 2005. "Controlled Stochastic Differential Equations under Poisson Uncertainty and with Unbounded Utility," Dresden Discussion Paper Series in Economics 03/05, Dresden University of Technology, Faculty of Business and Economics, Department of Economics.
  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 247-57, August.
  15. Walde, Klaus, 1999. "A Model of Creative Destruction with Undiversifiable Risk and Optimising Households," Economic Journal, Royal Economic Society, Royal Economic Society, vol. 109(454), pages C156-71, March.
  16. Steger, Thomas M., 2005. "Stochastic growth under Wiener and Poisson uncertainty," Economics Letters, Elsevier, Elsevier, vol. 86(3), pages 311-316, March.
  17. Aase, Knut Kristian, 1984. "Optimum portfolio diversification in a general continuous-time model," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 18(1), pages 81-98, September.
  18. Tjalling C. Koopmans, 1963. "On the Concept of Optimal Economic Growth," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 163, Cowles Foundation for Research in Economics, Yale University.
  19. Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, Elsevier, vol. 87(1), pages 194-217, July.
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Cited by:
  1. 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.
  2. 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.
  3. Klaus Wälde, 2009. "Production Technologies in Stochastic Continuous Time Models," CESifo Working Paper Series 2831, CESifo Group Munich.
  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, Department of Economics, UC San Diego qt1qp1g9ts, Department of Economics, UC San Diego.
  5. Levaggi, Rosella & Menoncin, Francesco, 2013. "Optimal dynamic tax evasion," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 37(11), pages 2157-2167.

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