# Martingale Functions of Brownian Motion and Its Local Time at 0

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Let $B = (B_t: t\ge 0)$ be a real-valued Brownian motion and let

$L = (L_t: t\ge 0)$ denote its local time in state 0. We present a characterization of the measurable functions $H$ such that $M_t = H(B_t,L_t)$

is a continuous local martingale. It turns out that the class of such functions is considerably wider when one relaxes the smoothness conditions that would be needed for a facile application of It\^o's formula.