  We study the construction of the 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
L\'{e}vy 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.
