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An example of a stochastic equilibrium with incomplete markets

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  • Gordan Zitkovic

Abstract

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.

Suggested Citation

  • Gordan Zitkovic, 2009. "An example of a stochastic equilibrium with incomplete markets," Papers 0906.0208, arXiv.org, revised Jun 2010.
  • Handle: RePEc:arx:papers:0906.0208
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    References listed on IDEAS

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    1. 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.
    2. 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, January.
    3. Darrell Duffie & Chi-Fu Huang, 2005. "Implementing Arrow-Debreu Equilibria By Continuous Trading Of Few Long-Lived Securities," World Scientific Book Chapters,in: Theory Of Valuation, chapter 4, pages 97-127 World Scientific Publishing Co. Pte. Ltd..
    4. 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 309-341.
    5. Duffie, Darrell & Zame, William, 1989. "The Consumption-Based Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 57(6), pages 1279-1297, November.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Frank Riedel & Peter Bank, 2001. "Existence and structure of stochastic equilibria with intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 487-509.
    11. 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.
    12. Herbert E. Scarf, 1967. "On the Computation of Equilibrium Prices," Cowles Foundation Discussion Papers 232, Cowles Foundation for Research in Economics, Yale University.
    13. Geanakoplos, John, 1990. "An introduction to general equilibrium with incomplete asset markets," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 1-38.
    14. Duffie, Darrell, 1986. "Stochastic Equilibria: Existence, Spanning Number, and the 'No Expected Financial Gain from Trade' Hypothesis," Econometrica, Econometric Society, vol. 54(5), pages 1161-1183, September.
    15. Kasper Larsen & Gordan Zitkovic, 2007. "Stability of utility-maximization in incomplete markets," Papers 0706.0474, arXiv.org.
    16. 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.
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