Modern Stochastics: Theory and Applications

In this note, we derive a large-deviation-type scaling limit for the discretization error of a Brownian stochastic integral. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.

We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter G-time-changed Brownian motions. In addition, we study the “sequence-independent variation” property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.