Is this a Brownian motion

Question

I am learning SDE, and here is some basic things I have trouble with,

Let $B(t)$ be a Brownian motion, and $F \in \mathcal L^2$ is any stochastic process and I know $\int_0^tF(s)dB(t)$ is Ito process so$\int_0^tF(s)dB(t)$ is a martingale, but is it a Brownian motion?

Answer 1

Compute variance of increment from time $t$ to time $s$ using Ito's isometry \begin{align} E[(\int_t^s F(u)dB(u))^2] = E(\int_t^s(F(u))^2du) = \int_t^sE(F(u))^2du \end{align}

In general, it's not compatible with the variance of Brownian increment, which is $s-t$.