Some ratio inequalities for iterated stochastic integrals

摘要

Let X = (X{sub}t, F{sub}t) be a continuous local martingale with quadratic variation 〈X〉 and X{sub}0 = 0. Define iterated stochastic integrals I{sub}n(X) = (I{sub}n(t, X),F{sub}t), n ≥ 0, inductively by I{sub}n(t, X) = ∫(I{sub}(n-1)(s, X)d)X{sub}s ) (X from 0 to t) with I{sub}0(t, X) = 1 and I{sub}1(t, X) = X{sub}t. Let (L{sub}t{sup}x(X)) be the local time of a continuous local martingale X at x ∈ R. Denote L{sub}t{sup}＊(X) = sup{sub}(x∈R)L{sub}t{sup}x(X) and X* = sup{sub}(t≥0) ｜X{sub}t｜. In this paper, we shall establish various ratio inequalities for I{sub}n (X}. In particular, we show that the inequalities c{sub}(n,p)‖(G(〈X〉{sub}∞)){sup}(n/2)‖{sub}p ≤ ‖sup{sub}(t≥0)｜I{sub}n(t,X)｜/(1＋〈X〉{sub}t){sup}(n/2)‖{sub}p ≤ C{sub}(n,p)‖(G(〈X〉{sub}∞)){sup}(n/2)‖{sub}p hold for 0 < p < ∞ with some positive constants c{sub}(n,p) and C{sub}(n,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[U{sub}n{sup}p exp(γ(U{sub}n{sup}(1))/V)] ≤ C{sub}(n,p,γ)E[V{sup}(np)](0<∞) holds with some positive constant C{sub}(n,p,γ) depending only on n, p and γ, where U{sub}n is one of 〈I{sub}n(X)〉{sub}∞{sup}(1/2) and I{sub}n{sup}＊(X), and V one of the three random variables X{sup}＊, 〈X〉{sub}∞{sup}(1/2) and L{sub}∞{sup}＊(X).