Stochastic equilibria in a general class of incomplete Brownian market environments
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This dissertation is a contribution to the equilibrium theory in incomplete financial markets. It shows that, under appropriate conditions, an equilibrium exists and is unique in a general class of incomplete Brownian market environments either composed of exponential-utility-maximizing agents or populated by a class of convex-risk-measure-minimizing agents. We first use the Dynamic Programming Principle to deduce the Hamilton-Jacobi-Bellman (HJB) equation for each agent, and solve the individual optimization problem, to identify the optimal control. Using the optimal portfolio, we establish the equivalence between the existence of a stochastic equilibrium in an incomplete Brownian market and solvability of a non-linearly coupled parabolic PDE system with a homogeneously-quadratic non-linear structure. To solve this PDE system, we work mainly in anisotropic Hölder spaces. There, we construct a proper class of Hölder subspaces, where potential solutions to the equilibrium PDE system are expected to “live”. These turn out to be convex and compact under the uniform topology, thanks to the help of an Arzela-Ascoli-type theorem for unbounded domains. We then define an approapriate functional on the subspace, and show that, if we choose the parameters associated with the subspace carefully, this functional maps the subspace back to itself. After that, we apply Schauder’s fixed point theorem on a constructed subset of the subspace, and establish the existence of solutions to the PDE system, therefore equivalently, the existence of market equilibria in these general incomplete Brownian market environments. To prove the uniqueness of the solution to the parabolic PDE system, we utilize classical L2-type energy estimates and the Gronwall’s inequality. This way, we also establish the uniqueness of a market equilibrium within a class of smooth Markovian markets.