Occupation times and beyond

Let Xt be a continuous local martingale satisfying X0=0 and K1q(t)[less-than-or-equals, slant] t[less-than-or-equals, slant]K2q(t) a.s. for a nondecreasing function q with constants K1 and K2. Define for a Borel function and Mt*=sup0[less-than-or-equals, slant]s[less-than-or-equals, slant]t Ms. If f is in L2 and f[not equal to]0 then for any slowly increasing function [phi] there exist two positive constants c and C such that for all stopping times TSuppose that f2 is even and is moderate. If [phi] satisfies one of the 3 conditions: (i) [phi] is slowly increasing, (ii) [phi] is concave if f[negated set membership]L2, and (iii) [phi] is moderate if is convex, then there exist two positive constants c and C such that for all stopping times TDefine Tr=inf {t>0; Mt=r}, r>0. The growth rate function of ETr[gamma] can be found for appropriate [gamma], 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.