On the Fields of Some Brownian Martingales

Abstract

Suppose $\{B_t\}_{t\geqq 0}$ is a standard 1-dimensional Brownian motion, and $f$ is a continuous function with nonaccumulating zero set. For $t \geqq 0$, let $M_t = \int^t_0 f(B_s) dB_s$. When does $M$ generate the same fields as $B$? When does $M$ generate the same fields as some Brownian motion? The answers to these questions are obtained; they involve the behavior of $f$ around its zeros. Also, either $M$ generates the same fields as some Brownian motion, or the fields of $M$ support discontinuous martingales.