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Controlled Stochastic Differential Equations under Poisson Uncertainty and with Unbounded Utility

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  • Sennewald, Ken
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    Abstract

    The present paper is concerned with the optimal control of stochastic differential equations, where uncertainty stems from one or more independent Poisson processes. Optimal behavior in such a setup (e.g., optimal consumption) is usually determined by employing the Hamilton-Jacobi-Bellman equation. This, however, requires strong assumptions on the model, such as a bounded utility function and bounded coefficients in the controlled differential equation. The present paper relaxes these assumptions. We show that one can still use the Hamilton-Jacobi-Bellman equation as a necessary criterion for optimality if the utility function and the coefficients are linearly bounded. We also derive sufficiency in a verification theorem without imposing any boundedness condition at all. It is finally shown that, under very mild assumptions, an optimal Markov control is optimal even within the class of general controls. --

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    Bibliographic Info

    Paper provided by Dresden University of Technology, Faculty of Business and Economics, Department of Economics in its series Dresden Discussion Paper Series in Economics with number 03/05.

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    Date of creation: 2005
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    Handle: RePEc:zbw:tuddps:0305

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    Related research

    Keywords: Stochastic differential equation; Poisson process; Bellman equation;

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    5. Michel, P., 1980. "On the Transversality Condition in Infinite Horizon Optimal Problems," Cahiers de recherche, Universite de Montreal, Departement de sciences economiques 8024, Universite de Montreal, Departement de sciences economiques.
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    11. Klaus, WAELDE, 2003. "Endogenous growth cycles," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales), Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES) 2004012, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES), revised 15 Mar 2004.
    12. 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.
    13. LE VAN, Cuong & MORHAIM, Lisa, 2001. "Optimal growth models with bounded or unbounded returns: a unifying approach," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 2001034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Kamihigashi, Takashi, 2001. "Necessity of Transversality Conditions for Infinite Horizon Problems," Econometrica, Econometric Society, Econometric Society, vol. 69(4), pages 995-1012, July.
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    Cited by:
    1. Olaf Posch & Klaus Wälde, 2006. "Natural volatility, welfare and taxation," Working Papers, Business School - Economics, University of Glasgow 2007_33, Business School - Economics, University of Glasgow.
    2. Sennewald, Ken & Wälde, Klaus, 2005. ""Ito's Lemma" and the Bellman equation for Poisson processes: An applied view," W.E.P. - Würzburg Economic Papers 58, University of Würzburg, Chair for Monetary Policy and International Economics.
    3. Sennewald, Ken & Wälde, Klaus, 2005. ""Itô's Lemma" and the Bellman equation: An applied view," Dresden Discussion Paper Series in Economics 04/05, Dresden University of Technology, Faculty of Business and Economics, Department of Economics.

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