Probability of exit from compact set

Question

I have a continuous real valued diffusion $\{ X_t \}_{t\ge0 }$ that is contained in a compact set $[a,b] $of $\mathbb{R}$, where $a > 0$ and. Define the stopping times \begin{equation} \tau_c=\inf \{ t \colon X_t \le c \}\qquad \text{ and } \qquad\tau_d=\inf \{ t \colon X_t \ge d \} \end{equation} I have by some calculation that for any $a<c<X_0<d<b$, $\mathbb{E}(\tau_c \land \tau_d)< \infty$. This means that $X_t$ exits from every compact subinterval of $(a,b)$ in finite expected time.

But why this implies $\mathbb{P}(\tau_c \land \tau_d < \infty)=1$? How can I show it?

Answer 1

Theorem Let $X$ be a nonnegative random variable. If $\mathbb E(X)<\infty$, then $\mathbb P(X<\infty)=1$.

Proof Assume on the contrary that $\mathbb P(X<\infty)<1$, or equivalently, $\mathbb P(X=\infty)>0$. Then $$ \mathbb E(X)\ge\infty\cdot P(X=\infty)=\infty, $$ a contradiction.
QED.