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The problem of model selection in the context of a system of stochasticdifferential equations (SDEs) has not been touched upon in the literature.Indeed, properties of Bayes factors have not been studied even in single SDEbased model comparison problems. In this article, we first develop anasymptotic theory of Bayes factors when two SDEs are compared, assuming thetime domain expands. Using this we then develop an asymptotic theory of Bayesfactors when systems of SDEs are compared, assuming that the number ofequations in each system, as well as the time domain, increase indefinitely.Our asymptotic theory covers situations when the observed processes associatedwith the SDEs are independently and identically distributed (iid), as well aswhen they are independently but not identically distributed (non-iid). Quiteimportantly, we allow inclusion of available time-dependent covariateinformation into each SDE through a multiplicative factor of the drift functionin a random effects set-up; different initial values for the SDEs are alsopermitted. Thus, our general model-selection framework includes simultaneouslythe variable selection problem associated with time-varying covariates, as wellas choice of the part of the drift function free of covariates. As we show, theBayes factor is inconsistent for comparing individual SDEs, in the sense thatthe log-Bayes factor converges only in expectation, while the relevant variancedoes not converge to zero. Nevertheless, it has been possible to exploit thisresult to establish almost sure exponential convergence of the Bayes factorwhen, in addition, the number of individuals are also allowed to increaseindefinitely. We carry out simulated and real data analyses to demonstrate thatBayes factor is a suitable candidate for covariate selection in our SDE modelseven in non-asymptotic situations.

Asymptotic Theory of Bayes Factor in Stochastic Differential Equations: Part II