Reflection principle

Remarks

Part 1 follows from the strong Markov property: after $\tau_a$, the process $W_{\tau_a + s} - a$ is a Wiener process independent of the past, and so is its negative. For part 2, the event $\{\sup_{s \le t} W_s \ge a\}$ equals $\{\tau_a \le t\}$, and it splits into $\{W_t \ge a\}$ and $\{\tau_a \le t, W_t < a\}$. By part 1 the second event has the same probability as $\{\widetilde{W}_t > a\} = \{W_t > a\}$, and $P(W_t = a) = 0$. As a consequence, $\sup_{s \le t} W_s$ has the same law as $\abs{W_t}$, and $P(\tau_a \le t) = 2\bigl(1 - \Phi(a/\sqrt{t})\bigr)$, where $\Phi$ is the standard normal distribution function. Differentiating gives the density $\frac{a}{\sqrt{2\pi t^3}} e^{-a^2/(2t)}$ of $\tau_a$, which shows $\tau_a < \infty$ almost surely but $E[\tau_a] = \infty$.

Going deeper

For a student

The principle gives the joint law of $W_t$ and its running maximum, and the law of first passage times. These are used to price barrier options and to study boundary crossing in sequential statistics. The same reflection idea for simple random walks gives the ballot theorem and counts of lattice paths.