  Model misspecification is a critical issue in many areas of theoretical and
empirical economics. In the specific context of misspecified Markov Decision
Processes, Esponda and Pouzo (2021) defined the notion of Berk-Nash equilibrium
and established its existence in the setting of finite state and action spaces.
However, many substantive applications (including two of the three motivating
examples presented by Esponda and Pouzo, as well as Gaussian and log-normal
distributions, and CARA, CRRA and mean-variance preferences) involve continuous
state or action spaces, and are thus not covered by the Esponda-Pouzo existence
theorem. We extend the existence of Berk-Nash equilibrium to compact action
spaces and sigma-compact state spaces, with possibly unbounded payoff
functions. A complication arises because the Berk-Nash equilibrium notion
depends critically on Radon-Nikodym derivatives, which are necessarily bounded
in the finite case but typically unbounded in misspecified continuous models.
The proofs rely on nonstandard analysis and, relative to previous applications
of nonstandard analysis in economic theory, draw on novel argumentation
traceable to work of the second author on nonstandard representations of Markov
processes.
