On a subclass of square integrable martingales

Abstract:

Let $\mathcal {M}_2^ \ast$ denote the class of continuous, nowhere constant, square integrable martingales, $M(t) = X({\langle M\rangle _t})$, for which ${\langle M\rangle _t}$ is a time change on the $\sigma$-fields generated by the Brownian motion $X(t)$. It is shown that if $M(t) \in \mathcal {M}_2^ \ast$, then the family of $\sigma$-fields generated by $M(t)$ is a right continuous family. If $M(t) \in \mathcal {M}_2^ \ast$ and if $\sigma \{ M(s):s \leqq t\} = \sigma \{ X(s):s \leqq t\}$ for some Brownian motion $X(t)$, then $M(t) = \smallint _0^t {\Phi (s)dX(s)}$ and $X(t) = \smallint _0^t {(1/\Phi (s))dM(s)}$ for some process $\Phi (s)$ with $\Phi (s) \ne 0$ a.e. $dt \times dP$.