Let $g:\mathbb{R^+} \to \mathbb{R^+}$ be a non decreasing and continuous function . Show that there exists a continuous martingale M such that its square bracket $<M>_t=g(t)-g(0)?)$
I have spent a few hours thinking about this question is a practice problem from my course on Stochastic calculus. I tried to consider the "martingale" $M_t=B_{g(t)}$ where B is a $1$-dim B.M which unfortunately did not work $M_t$ was not adapted.
The only continuous martingale which I can think of and play around with is a B.M. Any ideas as to how should I approach this problem? Even a hint would be appreciated. Thank you
Hint: Fix some square-integrable function $h$ and compute the square bracket of the process $X$ defined by $$X_t=\int_0^th(s)\mathrm dB_s.$$