Continuity of a stochastic process
Exercise 3.11 in Oksendal's book "SDEs: an introduction with applications".
Let $W_t$ be a stochastic process satisfying
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${W_t}$ is stationary;
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$t_1\not=t_2\implies W_{t_1}$ and $W_{t_2}$ are independent;
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$E[W_t]=0$ $\forall t$.
How to prove that $W_t$ cannot have continuous paths by considering $E[(W^{(N)}_t-W^{(N)}_s)^2]$ where $W^{(N)}_t=(-N)\vee(N\wedge W_t)$?
I am not sure where even where to start? We clearly have that $|W^{(N)}_t|\le N$ so maybe I should use bounded convergence?
If the paths are continuous then, for fixed $t$, you have $(W_s^{(N)} -W_t^{(N)})^2 \to 0$ when $s\to t$, and, by dominated convergence, the expectation goes to zero when $s\to t$, for every fixed $N$. On the other hand, $ E((W_s^{(N)} -W_t^{(N)})^2)= Var(W_s^{(N)}) +Var(W_t^{(N)}) \ge Var(W_t^{(N)})$, which, if $W_t$ is not degenerate, is bounded from zero when $N$ is large enough. So, if the paths are continuous and condition 2) holds, then every $W_t$ is degenerate. Then condition 3) implies that $W_t=0$ almost surely, for every $t$ (which indeed satisfies all the conditions). Condition 1) is superfetatory.