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We study the construction of the theoretical foundation of model comparisonfor ergodic stochastic differential equation (SDE) models and an extension ofthe applicable scope of the conventional Bayesian information criterion.Different from previous studies, we suppose that the candidate models arepossibly misspecified models, and we consider both Wiener and a pure-jumpL\'{e}vy noise driven SDE. Based on the asymptotic behavior of the marginalquasi-log likelihood, the Schwarz type statistics and stepwise model selectionprocedure are proposed. We also prove the model selection consistency of theproposed statistics with respect to an optimal model. We conduct some numericalexperiments and they support our theoretical findings.

Schwartz type model selection for ergodic stochastic differential equation models