Occupation times and beyond

Let Xt be a continuous local martingale satisfying X0=0 and K1q(t)⩽〈X〉t⩽K2q(t) a.s. for a nondecreasing function q with constants K1 and K2. Define for a Borel function f, Mt=∫0tf(Xs) dXs and Mt∗=sup0⩽s⩽t |Ms|. If f is in L2 and f≠0 then for any slowly increasing function φ there exist two positive constants c and C such that for all stopping times TcEφ(MT∗2)⩽Eφ(q(T))⩽C(Eφ(MT∗2)+1).Suppose that f2 is even and ψ(x)=∫0x(∫0yf2(t) dt) dy is moderate. If φ satisfies one of the 3 conditions: (i) φ is slowly increasing, (ii) φ is concave if f∉L2, and (iii) φ is moderate if ψ(x) is convex, then there exist two positive constants c and C such that for all stopping times TcEφ(MT∗2)⩽Eφ∘ψ(q(T))⩽CEφ(MT∗2).Define Tr=inf {t>0; |Mt|=r}, r>0. The growth rate function of ETrγ can be found for appropriate γ, as an application of the above inequalities. The method of proving the main result also yields a similar type of two-sided inequality for the integrable Brownian continuous additive functional over all stopping times.