On the distance between (X) and L∞ in the space of continuous BMO-martingales

Let X = (X-t,F-t) be a continuous BMO-martingale, that is, vertical bar vertical bar X vertical bar vertical bar(BMO)equivalent to(sup)(Tw) vertical bar vertical bar E[vertical bar X infinity - X-T vertical bar vertical bar FT]vertical bar vertical bar infinity < infinity , T where the supremum is taken over all stopping times T. Define the critical exponent b(X) by b(X) = {b > 0 : (sup)(T) vertical bar vertical bar E[exp(b(2)( X infinity - X T))vertical bar FT]vertical bar vertical bar infinity < infinity }, where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by q(X)(t) = E[(X)infinity vertical bar F-t] - E[ X infinity vertical bar F-0]. We use q(X) to characterize the distance between X and the class L-infinity of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities 4d1(q(X), L infinity) / 1 (<= b(X) <=) d1 (q(X), L infinity) / 4 hold for every continuous BMO-martingale X.