Showing a Quadratic Variation is increasing and continuous with respect to time

Question

Given $M$ a martingale starting from $0$ with continuous sample paths and $(M_t)_{t\ge 0}$ is a Gasussian process. I want to show $\langle M,M\rangle_t$ (the quadratic variation of $M_t$) is a monotone nondecreasing continuous function of $t$).

I was considering using Doob-Meyer. By Doob-Meyer there exists a nondecreasing stochastic process $(A_t)$ with continuous sample paths such that $A_t=_t.$

Is there a different way to proceed by using the fact that $M_t$ is Gaussian?

Answer 1

In fact (and maybe this is what you are saying you want to show), the quadratic variation of $M$ is deterministic; more precisely, $\langle M\rangle_t = E[M^2_t]$ a.s. Here is the outline of a proof. The function $q(s):=E[M_s^2]$ is continuous and non-decreasing. A Gaussian martingale necessarily has independent increments. Fix $t>0$. For a given positive integer $n$ choose $0=t_0