Brownian Motion Hitting Time and Reflection Principle

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Continuous Stochastic Processes / Reflection Principle.

Your Task
Let $B_t$ be a standard one-dimensional Brownian motion with $B_0 = 0$. For a fixed barrier $a > 0$, define the first hitting time $\tau_a = \inf \{ t \ge 0 : B_t = a \}$.
(1) Using the Bachelier-Lévy Reflection Principle, find the exact cumulative distribution function $\mathbb{P}(\tau_a \le t)$ and probability density function $f_{\tau_a}(t)$.
(2) Show that $\tau_a$ has a Lévy distribution and that $\mathbb{E}[\tau_a] = \infty$.

Constraints

• Transitions and increments satisfy the Markov or martingale property with respect to natural filtration.
• Stopping times are almost surely finite.