I am trying to prove the continuity of Ornstein-Uhlenbeck process which is a stationary Gaussian process with covariance kernel $k(x,y) = \exp(-|x-y])$.
Let $(X_t)_{t\ge 0}$ be an Ornstein-Uhlenbeck process with $0$ mean. The increment $(X_{t+h} – X_t)$ follows a Normal distribution with mean $0$ and variance $2(1-\exp(-h))$. In order to prove that the process is continuous, I need to show
$$ \Bbb{P}(\sup_{h<\delta} |X_{t+h} – X_t|> \epsilon)\xrightarrow[\delta \to 0]{} 0$$
Edit 1
Alternatively, one can show $$P(\lim_{h \to 0} X_{t+h} = X_t) = 1$$
It is enough to show that
\begin{equation}
P(\lim_{n \to \infty} X_{t+1/n} = X_t) = 1
\end{equation}
Now, $\sum_{n=1}^\infty P(|X_{t+1/n}-X_t|> \epsilon) < \sum_{n=1}^\infty \frac{var(X_{t+1/n}-X_t)}{\epsilon^2} = \sum_{n=1}^\infty \frac{(1-e^{-1/n})}{\epsilon^2}$
Hence, if we can bound the last term, then by Borel-Cantelli's lemma, the desired result will hold.
Are the details of the proof correct?
Best Answer
It is not enought to check continuity for each $t$. That is, the proof in the Edit 1 part is incorrect for two reasons,
1) It does not provide the continuity result, as you can see by a counter example. The poisson point process satisfies the condition $$ P(\lim_{h \to 0} X_{t+h} = X_t) = 1$$ and it is discontinuous almost surely (the reason this fails to prove continuity is that you have to consider non denumerable sets, one for each $t$)
2) The term
$$\sum_{n=1}^\infty P(|X_{t+1/n}-X_t|> \epsilon) < \sum_{n=1}^\infty \frac{var(X_{t+1/n}-X_t)}{\epsilon^2} = \sum_{n=1}^\infty \frac{(1-e^{-1/n})}{\epsilon^2} = \infty$$
So you can't apply Borel Cantelli.
Continuity for the Orstein-Uhlembeck(O-U) must be proved by other means. For instance, you could use Kolmogorov lemma to check if you can find a modification that has the same finite dimensionals and is continuous almost surely. Or you could find a sequence of random variables that converge to the O-U and satisfy the continuity condition.
$$ sup_n P(\sup_{h<\delta} |X^n_{t+h} - X^n_t|> \epsilon)\xrightarrow[\delta \to 0]{} 0$$
Calculon answer is also very good. Once you construct the O-U via Stochastic integration you immediately obtain continuous processes.
Ps: take a look at https://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem