Existence, Uniqueness, and Determinacy of Equilibria in Complete Security Markets with Infinite Dimensional Martingale Generator
2002-12
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Existence, Uniqueness, and Determinacy of Equilibria in Complete Security Markets with Infinite Dimensional Martingale Generator
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2002-12
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Center for Economic Research, Department of Economics, University of Minnesota
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Working Paper
Abstract
There is a strong evidence that most of financial variables are better
described by a combination of diffusion and jump processes. Considering such evidence,
researchers have studied security market models with jumps, in particular,
in the context of option pricing. In most of their models, jump magnitude is specified
as a continuously distributed random variable at each jump time. Then, the
dimensionality of martingale generator, which can be interpreted as the "number
of sources of uncertainty" in markets is infinite, and no finite set of securities can
complete markets. In security market economy with infinite dimensional martingale
generator, no equilibrium analysis has been conducted thus far. We assume
approximately complete markets (Bjork et al. [10] [11]) in which a continuum of
bonds are traded and any contingent claim can be approximately replicated with
an arbitrary precision. We introduce the notion of approximate security market
equilibrium in which an agent is allowed to choose a consumption plan approximately
supported with any prescribed precision. We prove that an approximate
security market equilibrium in approximately complete markets can be identified
with an Arrow-Debreu equilibrium. Then, we present sufficient conditions for the
existence of equilibria in the case of stochastic differential utilities with Inada condition,
and for the existence, uniqueness, and determinacy of equilibria in the case
of additively separable utilities.
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Previously Published Citation
Kusuda, K., (2002), "Existence, Uniqueness, and Determinacy of Equilibria in Complete Security Markets with Infinite Dimensional Martingale Generator", Discussion Paper No. 316, Center for Economic Research, Department of Economics, University of Minnesota.
Suggested citation
Kusuda, Koji. (2002). Existence, Uniqueness, and Determinacy of Equilibria in Complete Security Markets with Infinite Dimensional Martingale Generator. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/55883.
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