Maximal inequalities for the Ornstein-Uhlenbeck process

Abstract

Let V = (Vt)t≥ 0 be the Ornstein-Uhlenbeck velocity process solving d Vt = -β Vt dt + dBt with V0 = 0, where $\beta > 0$ and B = (Bt)t≥ 0 is a standard Brownian motion. Then there exist universal constants $C_1 > 0$ and $C_2 > 0$ such that $C_1\sqrt \beta E log (1 +\beta \tau) \leq E \big(0\leq t \leq \taumax |V_t|\big) \leq \frac C_2\sqrt\beta E log (1 + \beta \tau$ for all stopping times τ of V. In particular, this yields the existence of universal constants $D_1 > 0$ and $D_2 > 0$ such that $D_1 Elog(1 + log(1 + \tau)) \leq E\bigg(0\leq t\leq \taumax|B_t|1 + t \bigg)\leq D_2 Elog(1 + log(1 + \tau))$ for all stopping times τ 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 Ito calculus.