Model misspecification is a critical issue in many areas of economics. In the context of misspecified Markov Decision Processes, defined the notion of Berk-Nash equilibrium and established its existence with finite state and action spaces. However, many substantive applications (including two of the three motivating examples presented by Esponda and Pouzo) 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 utility functions. A complication arises because Berk-Nash equilibrium depends critically on Radon-Nikodym derivatives, which are bounded in the finite case but typically unbounded in misspecified continuous models. The proofs rely on nonstandard analysis, and draw on novel argumentation traceable to work of the second author on nonstandard representations of Markov processes.

中文翻译：

---

无限空间错误指定马尔可夫决策过程中 Berk-Nash 均衡的存在性

模型错误指定是许多经济学领域的一个关键问题。在错误指定的马尔可夫决策过程的背景下，定义了伯克-纳什均衡的概念，并在有限状态和行动空间中建立了其存在性。然而，许多实质性应用（包括 Esponda 和 Pouzo 提出的三个激励示例中的两个）涉及连续状态或动作空间，因此不被 Esponda-Pouzo 存在定理涵盖。我们将 Berk-Nash 均衡的存在性扩展到紧致行动空间和 sigma 紧致状态空间，并具有可能无界的效用函数。由于 Berk-Nash 平衡关键取决于 Radon-Nikodym 导数，这些导数在有限情况下有界，但在错误指定的连续模型中通常无界，因此出现了复杂情况。这些证明依赖于非标准分析，并利用可追溯到第二作者关于马尔可夫过程非标准表示的工作的新颖论证。