Bayes factors for a test about the drift of the Brownian motion under noninformative priors

Brownian motions are useful in modeling many stochastic phenomena. We address the problem of default testing for the sign of the drift, if any, in the mean of the process using the Bayesian approach. Conventional Bayes factors for hypotheses testing, however, cannot typically accommodate the use of standard noninformative priors, as such priors are defined only up to arbitrary constants which affect the values of the Bayes factors. To address this problem for some common noninformative priors, we shall use Intrinsic Bayes factors due to Berger and Pericchi (1996, J. Amer. Statist. Assoc. 91, 109-122) and fractional Bayes factors due to O'Hagan (1995, J. Roy. Statist. Soc. Ser. B 57(1), 99-138), assuming discrete observations are available from the process on a coarse time scale.