Maximal inequalities for the iterated fractional integrals

Let B-H(t) > 0 be a fractional Brownian motion with Hurst index 0 < H < 1. Define the iterated fractional integrals I-n(t, H), t greater than or equal to 0 inductively by I-n(t, H) = integral(0)(t) I-n 1 (s, H) dB(s)(H) with I-0(t, H) = I and I-t(t, H) = B-t(H). Then the inequalities [GRAPHICS] are proved to hold for all stopping times T of B-H and 0 < p less than or equal to infinity, where C-n,C-p and C-n,C-p are two positive constants depending only on n,p. As a related problem, we also obtain some moment inequalities for the process X-t = integral(0)(t) u(s) dB(s)(H) t greater than or equal to 0, where u is a deterministic continuous function with some suitable conditions. We show that [GRAPHICS] are equivalent for all 0 < p < infinity and all stopping times tau of B-H. (C) 2004 Elsevier B.V. All rights reserved.