Convergent martingales of asymptotically minimal fluctuation (Q1085886)

This paper establishes that certain martingales converge to their limits as fast as any other similarly adapted process, not just in a least squares sense, but also in a strong ''almost sure'' sense. In particular, a Brownian motion \((W_ t)\), stopped at a type of ''differentiably predictable'' stopping time T, possesses a minimal fluctuation property of the following sort: for any process \((Y_ t)\) adapted to the filtration generated by \((W_ t)\), \[ \overline{\lim}_{h\downarrow 0}[(2h \log \log h^{-1})^{-1/2}| W_ T-Y_{T-h}|]\geq 1\quad a.s., \] with equality if \(Y_ t=W_ t\). A random time-change argument extends this result to a class of continuous martingales; for instance, if \(M_ t\) is an Ito integral \(\int ^{t}_{0}g_ sdW_ s\) and \(Y_ t\) is a similarly adapted process, then \[ \overline{\lim}_{t\uparrow 1}[(2(1- t)\log \log (1-t)^{-1})^{-1/2}| M_ 1-Y_ t|]\geq | g_ 1| \quad a.s. \] with equality if \(Y_ t=M_ t\), provided that \(g^ 2_ t\) is continuous and positive at \(t=1\). Finally, discrete parameter martingales, obeying conditions required in Heyde's law of the iterated logarithm for the tails of convergent martingales, possess a minimal fluctuation property that can be interpreted as showing that Heyde's law gives a ''best'' rate of convergence.