Maximal inequalities for the Ornstein-Uhlenbeck process

Let $V=(V_t)_{t \ge 0}$be the Ornstein-Uhlenbeck velocity process solving

with $V_0=0$ , where $\beta >0$ and $B=(B_t)_{t \ge 0}$ is a standard Brownian motion. Then there exist universal constants $C_1>0$and $C_2>0$ such that

for all stopping times $\tau$ of $V$ . In particular, this yields the existence of universal constants $D_1>0$ and $D_2>0$ such that

for all stopping times $\tau$ of $B$. This inequality may be viewed as a stopped law of iterated logarithm. The method of proof relies upon a variant of Lenglart's domination principle and makes use of Itô calculus.