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An example of a stochastic equilibrium with incomplete markets

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  • Gordan Zitkovic
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    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.

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    File URL: http://arxiv.org/pdf/0906.0208
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0906.0208.

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    Date of creation: Jun 2009
    Date of revision: Jun 2010
    Handle: RePEc:arx:papers:0906.0208

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    1. 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.
    2. Clotilde Napp & Elyès Jouini, 2004. "Convergence of utility functions and convergence of optimal strategies," Post-Print halshs-00151579, HAL.
    3. Duffie, Darrell & Zame, William, 1989. "The Consumption-Based Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 57(6), pages 1279-97, November.
    4. Kasper Larsen & Gordan Zitkovic, 2007. "Stability of utility-maximization in incomplete markets," Papers 0706.0474, arXiv.org.
    5. Frank Riedel & Peter Bank, 2001. "Existence and structure of stochastic equilibria with intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 487-509.
    6. 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.
    7. Herbert E. Scarf, 1967. "On the Computation of Equilibrium Prices," Cowles Foundation Discussion Papers 232, Cowles Foundation for Research in Economics, Yale University.
    8. 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.
    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. 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.
    11. Geanakoplos, John, 1990. "An introduction to general equilibrium with incomplete asset markets," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 1-38.
    12. 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.
    13. 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.
    14. 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.
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