Some ratio inequalities for iterated stochastic integrals

Let X=(X-t,F-t) be a continuous local martingale with quadratic variation <X> and X-0=0. Define iterated stochastic integrals I-n (X)=(I-n (t, X), F-t), ngreater than or equal to0, inductively by I-n(t,X)=integral(0)(t) In-1(s,X) dX(s) with I-0 (t,X)=1 and I-1 (t,X)=X-t. Let (L-t(x) (X)) be the local time of a continuous local martingale X at x is an element of R. Denote L*(t) (X)=sup(xis an element ofR) L-t(x) (X) and X*=sup(tgreater than or equal to0) \X-t\. In this paper, we shall establish various ratio inequalities for I-n (X). In particular, we show that the inequalities [GRAPHICS hold for 0<p<infinity with some positive constants c(n,p) and C-n,C-P depending only on n and p, where G(t)=log(1+log(1+t)). Furthermore, we also show that for some gammagreater than or equal to0 the inequality [GRAPHICS] holds with some positive constant C-n,C-pgamma depending only on n, p and gamma, where U-n is one of <I-n(X)>(1/2)(infinity) I*(n) (X), and V one of the three random variables X*, (X)(infinity)(1/2) and L*(infinity) (X). (C) 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.