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

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  • Ken Sennewald
  • Klaus Wälde

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.

Suggested Citation

  • Ken Sennewald & Klaus Wälde, 2006. ""Itô’s Lemma" and the Bellman Equation for Poisson Processes: An Applied View," CESifo Working Paper Series 1684, CESifo Group Munich.
  • Handle: RePEc:ces:ceswps:_1684
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    1. repec:eee:apmaco:v:276:y:2016:i:c:p:407-416 is not listed on IDEAS
    2. Lucas Bretschger & Alexandra Vinogradova, 2014. "Growth and Mitigation Policies with Uncertain Climate Damage," CESifo Working Paper Series 5085, CESifo Group Munich.
    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. Bauer, Christian, 2009. "The Reservation Wage under CARA and Limited Borrowing," Discussion Papers in Economics 10291, University of Munich, Department of Economics.

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    Keywords

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

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