This question is from Protter's Stochastic calculus Problem 4 chapter 2.
Let $f:\mathbb{R}_+ \rightarrow \mathbb{R}_+$ be non-decreasing and continuous. Show that there exists continuous martingale $M$ such that $[M,M]_t=f(t)$
I only know how to show this when $f$ is absolute continuous. Suppose $f$ is absolute continuous, there exists $g\in L^1$ such that $f=\int_0^t g\ dt$. Then define $M$ as ($B_t$ is a Brownian motion) \begin{align*} M = \sqrt{g}\cdot B_t= \int_0^t \sqrt{g}\ dB_t \end{align*} By the property of quadratic variation \begin{align*} [M,M] = \int_0^t g \ dt = f(t) \end{align*} However, for the general $f$, I am not sure how to proceed. Does anyone have any idea?
Thanks in advance!
