Let $\xi_{t}$, $t\in$ $\mathbb{R}$ be the process with independent increments and continuous variance with respect to $t$. Is it true that $\xi_{t}$ is mean-square continuous?
For fixed $t\in{\mathbb{R}}$ $E({\xi_{t+h}-\xi_t})^2=D({\xi_{t+h}-\xi_t})+E^2({\xi_{t+h}-\xi_t})$. To obtain a counterexample, we must find a process with these properties and a discontinuous mean with respect to $t$. Is it possible or $\xi_t$ is mean-square continuous?
Let $\xi_t=f(t)+X$ This process has independent increments and variance of $\xi_t$ is variance of $X$ for all $t$. But $E(\xi_{t+h}-\xi_t)^{2}=(f(t+s)-f(t))^{2}$. Take $f$ to be a discontinuous function.