Controlled Stochastic Differential Equations under Poisson Uncertainty and with Unbounded Utility
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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- Gene M. Grossman & Elhanan Helpman, 1989.
"Quality Ladders in the Theory of Growth,"
NBER Working Papers
3099, National Bureau of Economic Research, Inc.
- Moen, Espen R, 1997.
"Competitive Search Equilibrium,"
Journal of Political Economy,
University of Chicago Press, vol. 105(2), pages 385-411, April.
- Bassan, Bruno & Çinlar, Erhan & Scarsini, Marco, 1993.
"Stochastic comparisons of Itô processes,"
Stochastic Processes and their Applications,
Elsevier, vol. 45(1), pages 1-11, March.
- Bellamy, Nadine, 2001. "Wealth optimization in an incomplete market driven by a jump-diffusion process," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 259-287, April.
- LE VAN, Cuong & MORHAIM, Lisa, 2001.
"Optimal growth models with bounded or unbounded returns: a unifying approach,"
CORE Discussion Papers
2001034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Le Van, Cuong & Morhaim, Lisa, 2002. "Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach," Journal of Economic Theory, Elsevier, vol. 105(1), pages 158-187, July.
- Le Van, C. & Morhaim, L., 2000. "Optimal Growth Models with Bounded or Unbounded Returns : a Unifying Approach," Papiers d'Economie MathÃ©matique et Applications 2000.64, UniversitÃ© PanthÃ©on-Sorbonne (Paris 1).
- Klaus, WAELDE, 2003.
"Endogenous growth cycles,"
Discussion Papers (IRES - Institut de Recherches Economiques et Sociales)
2004012, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES), revised 15 Mar 2004.
- Kamihigashi, Takashi, 2001. "Necessity of Transversality Conditions for Infinite Horizon Problems," Econometrica, Econometric Society, vol. 69(4), pages 995-1012, July.
- Juan Pablo RincÛn-Zapatero & Carlos RodrÌguez-Palmero, 2003. "Existence and Uniqueness of Solutions to the Bellman Equation in the Unbounded Case," Econometrica, Econometric Society, vol. 71(5), pages 1519-1555, 09.
- 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.
- Segerstrom, Paul S, 1998. "Endogenous Growth without Scale Effects," American Economic Review, American Economic Association, vol. 88(5), pages 1290-1310, December.
- R. C. Merton, 1970.
"Optimum Consumption and Portfolio Rules in a Continuous-time Model,"
58, Massachusetts Institute of Technology (MIT), Department of Economics.
- Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
- Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, vol. 87(1), pages 194-217, July.
- Fabien Postel-Vinay, 2002. "The Dynamics of Technological Unemployment," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 43(3), pages 737-760, August.
- Aase, Knut Kristian, 1984. "Optimum portfolio diversification in a general continuous-time model," Stochastic Processes and their Applications, Elsevier, vol. 18(1), pages 81-98, September.
- Steger, Thomas M., 2005. "Stochastic growth under Wiener and Poisson uncertainty," Economics Letters, Elsevier, vol. 86(3), pages 311-316, March.
- Michel, Philippe, 1982.
"On the Transversality Condition in Infinite Horizon Optimal Problems,"
Econometric Society, vol. 50(4), pages 975-85, July.
- Michel, P., 1980. "On the Transversality Condition in Infinite Horizon Optimal Problems," Cahiers de recherche 8024, Universite de Montreal, Departement de sciences economiques.
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