Exit Probability for Ornstein-Uhlenbeck Process

Let $(W_t)_{t \geq 0}$ be a standard Brownian motion, and let $X_t$ be defined by:

dX_t = -aX_t \, dt + \sigma \, dW_t, \quad X_0 = x \in (b, c), \quad a > 0,\; \sigma > 0,\; b \leq 0 \leq c.

Define the stopping time:

T := \inf\{t \geq 0 \mid X_t \notin (b, c)\}.

We admit that $\mathbb{P}(T < \infty) = 1$ .

Question: Determine the probability that $X_t$ exits the interval from the left, i.e.:

\mathbb{P}(X_T = b).

Answer