We study theoretical foundation of model comparison for ergodic stochastic differential equation (SDE) models and an extension of the applicable scope of the conventional Bayesian information criterion. Different from previous studies, we suppose that the candidate models are possibly misspecified models, and we consider both Wiener and a pure-jump Lévy noise-driven SDE. Based on the asymptotic behavior of the marginal quasi-log likelihood, the Schwarz-type statistics and stepwise model selection procedure are proposed. We also prove the model selection consistency of the proposed statistics with respect to an optimal model. We conduct some numerical experiments and they support our theoretical findings.

中文翻译：

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遍历随机微分方程模型的 Schwartz 型模型选择

我们研究遍历随机微分方程 (SDE) 模型的模型比较的理论基础和常规贝叶斯信息准则的适用范围的扩展。与之前的研究不同，我们假设候选模型可能是错误指定的模型，我们同时考虑了 Wiener 和纯跳跃 Lévy 噪声驱动的 SDE。基于边际拟对数似然的渐近行为，提出了 Schwarz 型统计量和逐步模型选择程序。我们还证明了所提出的统计量相对于最佳模型的模型选择一致性。我们进行了一些数值实验，它们支持了我们的理论发现。