Uniform in time modulus of continuity of Brownian motion

Abstract

Let $B=(B_t{)}_{t\ge 0}$ be a standard Brownian motion. The main objective is to find a uniform (in time) control of the modulus of continuity of $B$ in the spirit of what appears in (Kurtz, 1978). More precisely, it involves the control of the exponential moments of the random variable ${\sup }_{0\le s\le t}|B_t-B_s|/w(t,|t-s|)$ for a suitable function $w$. A stability inequality for diffusion processes is then derived and applied to two simple frameworks.