Consider the Ito equation:
$dX_t = f(t, X_t) dt + G(t, X_t) dW_t$
where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$, $G:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^{n\times m}$, $X_t \in \mathbb{R}^n$ and $W_t$ is an $m$-dimensional Wiener process.
Suppose that rank $G(t, x) \geq 1$ for all $t, x$. Then is it true that var$(X_t)$ is strictly increasing? If not, is it at least non-decreasing?
My thoughts on the 1-D case: Assume first for simplicity that $X_0=0$. Then in the case where both $X_t$ and $W_t$ are one-dimensional, we have
$X_t = \int_0^t f(s,X_s)ds + \int_0^t G(t,X_s)dW_s$
and since $X_t$ is an adapted process, we have
$E[X_t] = \int_0^t E[f(s,X_s)]ds$
Also, we have
$E[X_t^2] = E\left[\left(\int_0^tf(s,X_s)ds\right)^2\right]+ E\left[\int_0^t f(s,X_s)ds\int_0^t G(t,X_s)dW_s\right] + \int_0^t E[G(s,X_s)^2]ds$
where in the last term I have used the Ito isometry. Hence
var$[X_t]=E[X_t^2]-(E[X_t])^2 =$ var$\left[\int_0^t f(s,X_s)ds\right]+E\left[\int_0^t f(s,X_s)ds\int_0^t G(t,X_s)dW_s\right]$
$+\int_0^t E[G(s,X_s)^2]ds$
The first term is nonnegative and the last term is strictly increasing, since the 1-D version of my hypotheses is that $G(t,X_t)\not = 0, \, \forall t$. However, I don't know what I can say about the middle term.