General solvability of the mean field BSDE (3.1.2)

Establish the general solvability of the mean field backward stochastic differential equation (3.1.2) that characterizes the market-clearing equilibrium in the heterogeneous-agent asset-pricing model with habit formation, beyond the restrictive smallness assumptions used for existence in this paper.

Background

The paper derives a quadratic-growth mean field backward stochastic differential equation (BSDE), labeled (3.1.2), to endogenously construct the market risk premium under a market-clearing condition in an incomplete-market asset-pricing model with consumption habit formation. Existence and uniqueness of bounded solutions are proved using techniques that require additional smallness conditions on model parameters.

While these results establish solvability under specific assumptions, the authors highlight that obtaining general solvability for the BSDE (3.1.2) remains unresolved. This open problem is central because the mean field BSDE is the key mathematical object determining equilibrium in the large population limit, and broader conditions would extend the model’s applicability and robustness.