Small-time functional limit theorems for Itô diffusion processes

The standard small-time functional central limit theorem of semimartingales has been established in [Gerhold et al., 2015], proving that the scaling limit law of a large class of stochastic processes in increasingly small time scales is that of a Brownian motion with a possibly non-trivial variance-covariance matrix. In this presentation we focus on the time-homogeneous diffusion processes described by Ito SDEs. Instead of the simple time scaling 1/n of [Gerhold et al., 2015] we may consider the scaled processes stopped at the first exit times from the balls of decreasing radius n^{−1/2} without scaling time itself. This is a non-trivial example of a sequence of processes which converges in the sense of finite-dimensional distributions over a dense subset of [0, inf), but it does not converge weakly in the sense of laws of càdlàg processes.