On Brownian motion as a prior for nonparametric regression

In this paper we consider the use of Brownian motion as a prior in a nonparametric, univariate regression setting. Using change of measure theory for continuous semimartingales we derive an explicit stochastic differential equation characterization for the posterior. In combination with stochastic calculus tools this dynamical characterization of the posterior allows us to derive new asymptotic properties of the posterior.