Let $(X_t, t \in \mathbb{R})$ be an Ornstein-Uhlenbeck process, i.e. $X_t$ is defined by $$X_t = \sigma \int_{-\infty}^t e^{-\theta(t-s)} dW_s$$ for $t \in \mathbb{R}$ for parameters $\theta, \sigma > 0$.
Show that for an arbitrary compact interval $T \subset \mathbb{R}$ there exists constants $p,q,C > 0$ such that $$\mathbb{E}|X_t - X_s|^p \leq C|t-s|^{1+q}$$ for all $s, t \in T$.
We want $q/p = 1/2$, so let $p=2$ and $q=1$, then $$\mathbb{E}|X_t - X_s|^2 = \mathbb{E}X_t^2 - 2\mathbb{E}(X_t X_s) + \mathbb{E}X_s^2 \\ = \frac{\sigma^2}{2\theta} - 2\frac{\sigma^2}{2\theta}e^{-\theta|t-s|} + \frac{\sigma^2}{2\theta} \\ = \frac{\sigma^2}{\theta}(1 - e^{-\theta|t-s|}).$$
This should be smaller than $C|t-s|^2$ for all $s,t \in T$ for some $C > 0$. I cannot find this $C$, since $\frac{\sigma^2}{\theta}(1 - e^{-\theta|t-s|})$ will always be above $C|t-s|^2$ if $t-s \rightarrow 0$. The bigger $C$ is chosen the 'earlier' the inequality will hold, but it will never hold when $t \rightarrow s$ (if I'm correct).
What am I doing wrong? Actually I want to prove that the Ornstein-Uhlenbeck process admits a version that is locally Hölder continuous of order strictly less than 1/2. I thought I should've made use of Kolmogorov's Continuity Criterion, but that doesn't seem to work out.
Does anyone know what I should do?
Thanks in advance!
Edit
I'm still having trouble when considering $p=4$ and $q=2$, because then I obtain $$\mathbb{E}|X_t - X_s|^4 = \frac{3\sigma^4}{\theta^2}\left(e^{-\theta|t-s|} - 1\right)^2$$ and I can't find a $C>0$ such that $$\frac{3\sigma^4}{\theta^2}\left(e^{-\theta|t-s|} - 1\right)^2 \leq C|t-s|^3$$ for all $s,t \in \mathbb{R}$.
What am I doing wrong?