A Counterexample to Small-time Limit Theorems for Stochastic Processes

The standard small-time functional central limit theorem of semimartingales has been established in (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics, 87), 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 nontrivial variance-covariance matrix. In this paper we focus on the time-homogeneous diffusion processes described by It\^{o} SDEs. Instead of the simple time scaling $1/n$ of (Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications. Stochastics, 87) we consider the scaled processes stopped at the first exit times from the balls of decreasing radius $n^{-1/2}$ without scaling time itself. To the best of our knowledge, this particular scaling has not been investigated in the literature. We prove that this is a nontrivial example of a sequence of processes which converges in the sense of finite-dimensional distributions over a dense subset of $[0,\infty)$, but it does not converge weakly in the sense of laws of c\`{a}dl\`{a}g processes. We also characterise the limit law of the scaled processes evaluated at their respective first exit times.