Abstract
In the setting of exponential investors and uncertainty governed by Brownian motions, we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential–quadratic models and prove that the corresponding incomplete equilibrium is characterized by a coupled set of Riccati equations. Finally, we prove that these exponential–quadratic models can be used to approximate the incomplete models we studied in the first part.
Highlights
In a multi-dimensional auto-regressive Brownian setting with heterogeneous exponential utility investors we first prove that an incomplete equilibrium exists
We construct a class of incomplete models for which the equilibrium is described by coupled Riccati equations
We show that this tractable class of models can be used as a Taylor approximation of the general class of models we first considered
Summary
In a multi-dimensional auto-regressive Brownian setting with heterogeneous exponential utility investors we first prove that an incomplete equilibrium exists. The existence of an incomplete equilibrium in the case of multiple Brownian motions is proven in the thesis [16] using Schauder’s fixed point theorem under a decay property of the endowments (see Assumption 2.3.1 in [16]). In the second part of this paper we construct a class of exponential-quadractic models for which the corresponding equilibrium is characterized by a coupled set of Riccati equations. We show that the general setting’s incomplete equilibrium can be approximated (for short time-horizons) by replacing the individual investors’ endowments with their second degree Taylor approximations. This type of approximation falls into the setting of exponential-quadratic models.
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