Section 18.2

Hitting Times and the Reflection Principle

Exit probabilities, the running maximum, and the distribution and moments of hitting times.

Toolkit

Hitting times and running maxima of Brownian motion rely on optional stopping (martingale) arguments and the reflection principle, which exploits the strong Markov property and symmetry of increments after a hitting time.

Key results, for standard $W_t$ started at $0$ , with $T_a=\inf\{t:W_t=a\}$ :

Gambler’s ruin (martingale OST): with $T=T_{-a}\wedge T_b$ ,
$\mathbb{P}(\text{hit } b \text{ before } -a)=\frac{a}{a+b},\qquad \mathbb{E}[T]=ab.$
Reflection principle: for $M_t=\max_{s\le t}W_s$ ,
$\mathbb{P}(M_t\ge a)=2\,\mathbb{P}(W_t\ge a)=\mathbb{P}(|W_t|\ge a),\qquad a\ge0,$
and $M_t\overset{d}{=}|W_t|$ .
Hitting time density: $T_a$ has density
$f_{T_a}(t)=\frac{a}{\sqrt{2\pi t^3}}e^{-a^2/2t},\qquad a>0,$
an a.s. finite but $\mathbb{E}[T_a]=\infty$ random variable.
Laplace transform via exponential martingale $\exp(\theta W_t-\theta^2t/2)$ :
$\mathbb{E}\big[e^{-\lambda T_a}\big]=e^{-a\sqrt{2\lambda}},\qquad \mathbb{E}\big[e^{-\lambda T}\big]=\frac{1}{\cosh(a\sqrt{2\lambda})}\ \text{for symmetric two-sided exit at }\pm a.$

Problems