Martingale Functions of Brownian Motion and its Local Time

P.J. Fitzsimmons, D.M. Wroblewski

2010, v.16, №3, 599-608

ABSTRACT

We characterize the class of local martingales of the form $H(B_t,L_t)$ for a standard one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$ and its local time at $0$, $L=(L_t)_{t\ge 0}$. The main result is closely related to work of J. Obloj, who studied the local martingales of the form $H(B_t,\ov B_t)$, where $\ov B_t = \sup_{0\le s\le t} B_s$.

Keywords: Brownian motion,local martingale,local time

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