A stopping time is given as $\tau = \text{inf}\lbrace t≥0 : X_t ≤ 0 \rbrace $ and the stochastic process is defined as
\begin{equation} X_t = (a-B_t)e^{B_t} \end{equation}
where $a>0$ and $B_t$ is a standard continuous Brownian motion ( w.r.t. its augmented natural filtration $\mathcal{G_t}= \sigma(B_s | s≤t)$ $\vee$ $\mathcal{N} $ )
In the exercise, I am asked to compute:
(1) $\mathbb{P} (\tau ≤ t)$ in terms of $a, t$ and $\Phi$ (the cdf of a standard normal).
(2) $\mathbb{P(\tau = + \infty)}$.
How would you tackle these types of problem? Can you help me solve this exercise? Thank you in advance.