INEQUALITIES FOR INCREMENTS OF STOCHASTIC-PROCESSES AND MODULI OF CONTINUITY

Let {GAMMA(t), t is-an-element-of R} be a Banach space B-valued stochastic process. Let P be the probability measure generated by GAMMA(.). Assume that GAMMA(.) is P-almost surely continuous with respect to the norm parallel-to parallel-to of B and that there exists a positive nondecreasing function sigma(a), a > 0, such that P{parallel-to GAMMA(t + a) - GAMMA(t)parallel-to greater-than-or-equal-to x-sigma(a)} less-than-or-equal-to K exp(-gamma-x(beta) with some K, gamma, beta > 0. Then, assuming also that sigma(.) is a regularly varying function at zero, or at infinity, with a positive exponent, we prove large deviation results for increments like sup0 less-than-or-equal-to t less-than-or-equal-to T-a sup0 less-than-or-equal-to s less-than-or-equal-to a parallel-to GAMMA(t + s) - GAMMA(t)parallel-to, which we then use to establish moduli of continuity and large increment estimates for GAMMA(.). One of the many applications is to prove moduli of continuity estimates for l2-valued Ornstein-Uhlenbeck processes.