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

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  • Ken Sennewald
  • Klaus Waelde

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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File URL: http://www.cesifo-group.de/portal/page/portal/DocBase_Content/WP/WP-CESifo_Working_Papers/wp-cesifo-2006/wp-cesifo-2006-03/cesifo1_wp1684.pdf
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Bibliographic Info

Paper provided by CESifo Group Munich in its series CESifo Working Paper Series with number 1684.

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Date of creation: 2006
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Handle: RePEc:ces:ceswps:_1684

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Keywords: stochastic differential equation; Poisson process; Bellman equation; portfolio optimization; consumption optimization;

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References

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  1. 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.
  2. 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.
  3. Moen, E.R., 1995. "Competitive Search Equilibrium," Memorandum, Oslo University, Department of Economics 37/1995, Oslo University, Department of Economics.
  4. Maurice Obstfeld., 1993. "Risk-Taking, Global Diversification, and Growth," Center for International and Development Economics Research (CIDER) Working Papers, University of California at Berkeley C93-016, University of California at Berkeley.
  5. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
  6. Aghion, P. & Howitt, P., 1989. "A Model Of Growth Through Creative Destruction," UWO Department of Economics Working Papers, University of Western Ontario, Department of Economics 8904, University of Western Ontario, Department of Economics.
  7. Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, Elsevier, vol. 87(1), pages 194-217, July.
  8. Chang,Fwu-Ranq, 2004. "Stochastic Optimization in Continuous Time," Cambridge Books, Cambridge University Press, Cambridge University Press, number 9780521834063.
  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. Grossman, Gene M & Helpman, Elhanan, 1991. "Quality Ladders in the Theory of Growth," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 58(1), pages 43-61, January.
  11. 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.
  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. 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.
  14. 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.
  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. Aghion, Philippe & Howitt, Peter, 1992. "A Model of Growth Through Creative Destruction," Scholarly Articles 12490578, Harvard University Department of Economics.
  17. 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.
  18. Steger, Thomas M., 2005. "Stochastic growth under Wiener and Poisson uncertainty," Economics Letters, Elsevier, Elsevier, vol. 86(3), pages 311-316, March.
  19. 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.
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Cited by:
  1. Levaggi, Rosella & Menoncin, Francesco, 2013. "Optimal dynamic tax evasion," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 37(11), pages 2157-2167.
  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. 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.
  5. 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.

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