Delattre et al. (2013) considered n independent stochastic differential equations (SDE's), where in each case the drift term is associated with a random effect, the distribution of which depends upon unknown parameters. Assuming the independent and identical (iid) situation the authors provide independent proofs of weak consistency and asymptotic normality of the maximum likelihood estimators (M LE's) of the hyper-parameters of their random effects parameters.In this article, as an alternative route to proving consistency and asymptotic normality in the SDE set-up involving random effects, we verify the regularity conditions required by existing relevant theorems. In particular, this approach allowed us to prove strong consistency under weaker assumption. But much more importantly, we further consider the independent, but non-identical set-up associated with the random effects based SDE framework, and prove asymptotic results associated with the M LE's.

Delattre et al. (2013) investigated asymptotic properties of the maximum likelihood estimator of the population parameters of the random effects associated with n independent stochastic differential equations (SDE's) assuming that the SDE's are independent and identical (iid).In this article, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution in the iid set-up as well as when the SDE's are independent but non-identical.

The problem of model selection in the context of a system of stochastic differential equations (SDE's) has not been touched upon in the literature. Indeed, properties of Bayes factors have not been studied even in single SDE based model comparison problems.In this article, we first develop an asymptotic theory of Bayes factors when two SDE's are compared, assuming the time domain expands. Using this we then develop an asymptotic theory of Bayes factors when systems of SDE's are compared, assuming that the number of equations in each system, as well as the time domain, increase indefinitely. Our asymptotic theory covers situations when the observed processes associated with the SDE's are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). Quite importantly, we allow inclusion of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function in a random effects set-up; different initial values for the SDE's are also permitted.Thus, our general model-selection framework includes simultaneously the variable selection problem associated with time-varying covariates, as well as choice of the part of the drift function free of covariates. It is to be noted that given that the underlying process is wholly observed, the diffusion coefficient becomes known, and hence is not involved in the model selection problem.For both iid and non-iid set-ups we establish almost sure exponential convergence of the Bayes factor. As we show, the Bayes factor is inconsistent for comparing individual SDE's, in the sense that the log-Bayes factor converges only in expectation, while the relevant variance does not converge to zero. Nevertheless, it has been possible to exploit this result to establish almost sure exponential convergence of the Bayes factor when, in addition, the number of individuals are also allowed to increase indefinitely.We carry out simulated and real data analyses to demonstrate that Bayes factor is a suitable candidate for covariate selection in our SDE models even in non-asymptotic situations.

Although there is a significant literature on the asymptotic theory of Bayes factor, the set-ups considered are usually specialized and often involves independent and identically distributed data. Even in such specialized cases, mostly weak consistency results are available. In this article, for the first time ever, we derive the almost sure convergence theory of Bayes factor in the general set-up that includes even dependent data and misspecified models. Somewhat surprisingly, the key to the proof of such a general theory is a simple application of a result of Shalizi (2009) to a well-known identity satisfied by the Bayes factor.

In this article, we introduce a system of stochastic differential equations (SDEs) consisting of time‐dependent covariates and consider both fixed and random effects. We also allow the functional part associated with the drift function to depend upon unknown parameters. For this general SDE system we establish consistency and asymptotic normality of the maximum likelihood estimator. We consider a Bayesian approach for learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution. We supplement our theoretical investigation with simulated and real data analyses, obtaining encouraging results in both cases. The Canadian Journal of Statistics 46: 635–655; 2018 © 2018 Statistical Society of Canada