| Abstract: | Let X = (Xt, ?t) be a continuous local martingale with quadratic variation 〈X〉 and X0 = 0. Define iterated stochastic integrals In(X) = (In(t, X), ?t), n ≥ 0, inductively by $$ I_{n} (t, X) = int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I0(t, X) = 1 and I1(t, X) = Xt. Let (??xt(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = supx∈? ??xt(X) and X* = supt≥0 |Xt|. In this paper, we shall establish various ratio inequalities for In(X). In particular, we show that the inequalities $$ c_{n,p} , leftVert (G ( langle X rangle _{infty} )) ^{n/2} rightVert _{p} ; le ; leftVert {mathop sup limits _{t ge 0}} ; {leftvert I_{n} (t, X) rightvert over {(1+ langle X rangle _{t} ) ^{n/2}}} rightVert _{p} ; le C_{n, p} , leftVert (G ( langle X rangle _{infty} )) ^{n/2} rightVert _{p} $$ hold for 0 < p < ∞ with some positive constants cn,p and Cn,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E left[ U ^{p}_{n} exp left( gamma {U ^{1/n} _{n} over {V}} right) right] le C_{n, p, gamma} E [V ^{n, p}] quad (0 < p < infty ) $$ holds with some positive constant Cn,p,γ depending only on n, p and γ, where Un is one of 〈In(X)〉1/2∞ and I*n(X), and V one of the three random variables X*, 〈X〉1/2∞ and ??*∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
