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"Itô's Lemma" and the Bellman equation: 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.

Suggested Citation

  • Sennewald, Ken & Wälde, Klaus, 2005. ""Itô's Lemma" and the Bellman equation: An applied view," Dresden Discussion Paper Series in Economics 04/05, Technische Universität Dresden, Faculty of Business and Economics, Department of Economics.
    • Handle: RePEc:zbw:tuddps:0405
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    References listed on IDEAS

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    More about this item

    Keywords

    Stochastic differential equation; Poisson process; Bellman equation; Portfolio optimization; Consump;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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