Stochastic-integral representation from absolutely continuous quadratic variation
= Stochastic-integral representation from absolutely continuous quadratic variation
If a continuous local martingale satisfies
$$
[M]_t=\int_0^tA_s\,ds
$$
for a nonnegative predictable process $A$, then, after enlarging the probability space if necessary, there is a Brownian motion $B$ such that
$$
M_t-M_0=\int_0^t\sqrt{A_s}\,dB_s.
$$
An independent Brownian motion supplies noise on the set where $A=0$.