An example of a stochastic equilibrium with incomplete markets
We prove existence and uniqueness of stochastic equilibria in a class of incomplete continuous-time financial environments where the market participants are exponential utility maximizers with heterogeneous risk-aversion coefficients and general Markovian random endowments. The incompleteness featured in our setting - the source of which can be thought of as a credit event or a catastrophe - is genuine in the sense that not only the prices, but also the family of replicable claims itself is determined as a part of the equilibrium. Consequently, equilibrium allocations are not necessarily Pareto optimal and the related representative-agent techniques cannot be used. Instead, we follow a novel route based on new stability results for a class of semilinear partial differential equations related to the Hamilton-Jacobi-Bellman equation for the agents' utility-maximization problems. This approach leads to a reformulation of the problem where the Banach fixed point theorem can be used not only to show existence and uniqueness, but also to provide a simple and efficient numerical procedure for its computation.
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- Bank, Peter & Riedel, Frank, 2000.
"Existence and structure of stochastic equilibria with intertemporal substitution,"
SFB 373 Discussion Papers
2000,104, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
- Frank Riedel & Peter Bank, 2001. "Existence and structure of stochastic equilibria with intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 487-509.
- Clotilde Napp & Elyès Jouini, 2004.
"Convergence of utility functions and convergence of optimal strategies,"
- Elyès Jouini & Clotilde Napp, 2004. "Convergence of utility functions and convergence of optimal strategies," Finance and Stochastics, Springer, vol. 8(1), pages 133-144, January.
- Duffie, J Darrell & Huang, Chi-fu, 1985. "Implementing Arrow-Debreu Equilibria by Continuous Trading of Few Long-lived Securities," Econometrica, Econometric Society, vol. 53(6), pages 1337-56, November.
- Kasper Larsen & Gordan Zitkovic, 2007. "Stability of utility-maximization in incomplete markets," Papers 0706.0474, arXiv.org.
- Damir Filipović & Michael Kupper, 2008. "Equilibrium Prices For Monetary Utility Functions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(03), pages 325-343.
- Magill, Michael & Shafer, Wayne, 1991. "Incomplete markets," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 30, pages 1523-1614 Elsevier.
- Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
- Darrell Duffie & William Zame, 1988.
"The Consumption-Based Capital Asset Pricing Model,"
88-10, University of Copenhagen. Department of Economics.
- Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123.
- Radner, Roy, 1972. "Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets," Econometrica, Econometric Society, vol. 40(2), pages 289-303, March.
- Gordan Žitković, 2006. "Financial equilibria in the semimartingale setting: Complete markets and markets with withdrawal constraints," Finance and Stochastics, Springer, vol. 10(1), pages 99-119, 01.
- Ioannis Karatzas & John P. Lehoczky & Steven E. Shreve, 1991. "Equilibrium Models With Singular Asset Prices," Mathematical Finance, Wiley Blackwell, vol. 1(3), pages 11-29.
- Duffie, Darrell, 1986. "Stochastic Equilibria: Existence, Spanning Number, and the 'No Expected Financial Gain from Trade' Hypothesis," Econometrica, Econometric Society, vol. 54(5), pages 1161-83, September.
- Larsen, Kasper & Zitkovic, Gordan, 2007. "Stability of utility-maximization in incomplete markets," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1642-1662, November.
- Basak, Suleyman & Cuoco, Domenico, 1998. "An Equilibrium Model with Restricted Stock Market Participation," Review of Financial Studies, Society for Financial Studies, vol. 11(2), pages 309-41.
- Herbert E. Scarf, 1967. "On the Computation of Equilibrium Prices," Cowles Foundation Discussion Papers 232, Cowles Foundation for Research in Economics, Yale University.
- Geanakoplos, John, 1990. "An introduction to general equilibrium with incomplete asset markets," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 1-38.
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