Let $X_t$ ($t \in \mathbb R$) be a random process with independent increments such that its variance $DX_t$ is continuous as a function of $t$. Is $X_t$ $L_2$-continuous?
It is well known that random process is $L_2$-continuous iff $\mathbb E X_t$ is continous as function of $t$ and its autocorrelation function $R(t, s)$ is continuous in the points of the form $(t, t)$ ($s = t$).
I was not able to deduce the later facts from the given information, so I assume that answer is not always. But I also can't find an example to prove it.
So any help is appreciated. Thanks!
UPD: So finally it seems like I found pretty simple example: let $X_0 = 1$ everywhere on probability space and $X_t = 0$ whenever $t \neq 0$. $D X_t = 0$ for all $t$ and thus varience is continuous. All increments are simply constant random variables and therefore independent. But process is $L_2$-discontinuous at 0 since $\operatorname{lim}_{t \rightarrow 0}\mathbb E(X_0 - X_t)^2 = 1$. Am I missing something?