Ornstein - Uhlenbeck - Stochastic Differential Equation liminf and limsup I came across an exercise which deals with a stochastic differential equation of the form
$$\mathrm dX(t)=-\theta[X(t)-\mu]\mathrm dt+\sigma\,\mathrm dW(t)$$
for $t>0$, where $\theta,\sigma>0$ and $\mu$ are fixed parameters. It requests to show that $$\lim \sup X(t)=\infty$$ almost certainly when $t$ approaches infinity. Also it requests to show that  $$\lim \inf X(t)=-\infty$$ almost certaintly as $t$ approaches infinity. It says to use the Dambis–Dubins–Schwarz theorem and the law of iterated logarithms for Brownian motion: $$ \lim \sup \frac{W(t)}{\sqrt{2t}\log\log t}=1\quad\text{and}\quad\lim \inf \frac{W(t)}{\sqrt{2t}\log\log t}=-1 $$ almost certainly as $t$ approaches infinity.
Any help on how I can solve/prove these? Thanks.
 A: The quadratic variation of the Ornstein-Uhlenbeck process above is
$$[X]_t = \sigma^2 [W]_t = \sigma^2 t,$$
which is unbounded. Moreover, the O.-U. process is a continuous local martingale, and we assume that $X_0 = 0$. We define the process $X^\tau = \{X_{\tau_t}, t \geq 0\}$ defined by the times
$$\tau_t = \inf\{s \geq 0 : [X]_s > t\}.$$
By the theorem of Dambis-Dubins-Schwarz, the process $X^\tau$ is indistinguishable from a Brownian motion up until time $[X]_\infty$. Since the process is divergent, the result applies for all $t \geq 0$. Since $X^\tau$ is a Brownian motion, the Law of Iterated Logarithm implies that, almost surely,
$$\limsup_{t\rightarrow\infty} X^\tau_t = \infty\tag{1}$$
and
$$\liminf_{t\rightarrow\infty} X^\tau_t = -\infty.\tag{2}$$
We can translate these results about the process $X^\tau$ into ones about the original process $X$, since any time we have $X^\tau_t > N$ for some large $N$, we necessarily have $X_{\tau_t} > N$; ditto for $X^\tau_t < -N$. Thus, $(1)$ and $(2)$ hold for $X_t$ in place of $X^\tau_t$, as was to be shown.
