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I thought there was a post asking the question in the title, but cannot find it anymore.

It is not difficult to construct a process with countable many continuous sample paths, all starting from 0, but has discontinous variance at 0.

So, I will add some additional conditions:

Let $X_t$ be a Markov process with continuous sample paths, prove (or give a counter example) that $Var(X_t)$ is continuous with respect to $t$

Also, how about the case where $X_t$ is a martingale instead of Markovian? Perhaps the time-changed Brownian motion theorem can be helpful?

[Edit] come to think of it, in the Markovian case, even the continuity of $E(X_t)$ is not obvious. Any known results?

Jay.H
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  • H : Unless mistaken essentially this is a question of L^1 or L^2 convergence, when you have pointwise convergence. Find any counterexample in the classical integration theory for (look at when the theorem dominated convergence of Lebesgue does not apply and try to adapt it in a process context (markovian or martingale). Best regards – TheBridge Dec 11 '15 at 15:16
  • We only ask for the continuity in $E(X_t) $ and $E(X_t^2)$, not in $X_t$ itself, so finding an counter example will be harder. Also, with the additional condition of Markov or Martingale, this does not seem trivial to me. – Jay.H Dec 11 '15 at 17:04

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