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Stochastic equilibria and stability in a class of incomplete continuous-time financial environments

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Author Info
Gordan Zitkovic
Abstract

We prove existence and uniqueness of stochastic equilibria in a representative class of incomplete continuous-time financial environments where the market participants are exponential utility maximizers with heterogeneous risk-aversion coefficients and general 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, the usual approach which employs the a-posteriori Pareto optimality of equilibrium allocations and the related representative-agent techniques in the complete-market setting 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/abs/0906.0208
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Paper provided by arXiv.org in its series Quantitative Finance Papers with number 0906.0208.

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

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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. [Downloadable!] (restricted)
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  2. Frank Riedel & Peter Bank, 2001. "Existence and structure of stochastic equilibria with intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 487-509. [Downloadable!] (restricted)
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  3. 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. [Downloadable!] (restricted)
  4. Herbert E. Scarf, 1967. "On the Computation of Equilibrium Prices," Cowles Foundation Discussion Papers 232, Cowles Foundation, Yale University. [Downloadable!]
  5. Basak, Suleyman & Cuoco, Domenico, 1998. "An Equilibrium Model with Restricted Stock Market Participation," Review of Financial Studies, Oxford University Press for Society for Financial Studies, vol. 11(2), pages 309-41.
  6. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation, Yale University. [Downloadable!]
  7. Geanakoplos, John, 1990. "An introduction to general equilibrium with incomplete asset markets," Journal of Mathematical Economics, Elsevier, vol. 19(1-2), pages 1-38. [Downloadable!] (restricted)
  8. 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. [Downloadable!] (restricted)
  9. 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. [Downloadable!] (restricted)
  10. 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. [Downloadable!] (restricted)
  11. Kasper Larsen & Gordan Zitkovic, 2007. "Stability of utility-maximization in incomplete markets," Quantitative Finance Papers 0706.0474, arXiv.org. [Downloadable!]
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