Strong Markov Property for Brownian Motion

The strong Markov property states that Brownian motion "restarts" at a stopping time, independent of the past.

Statement

Theorem4.1Strong Markov property

Let $\tau$ be a stopping time with $\mathbb{P}(\tau < \infty) = 1$ . Define the shifted process $\tilde{B}_t = B_{\tau+t} - B_\tau$ for $t \geq 0$ . Then $(\tilde{B}_t)$ is a Brownian motion independent of $\mathcal{F}_\tau = \sigma(B_s : s \leq \tau)$ .

This extends the ordinary Markov property (which holds at fixed times) to random times determined by the history of the process.

ExampleDistribution after hitting

Let $\tau_a = \inf\{t : B_t = a\}$ . By the strong Markov property, $(B_{\tau_a + t} - a)$ is a Brownian motion starting at 0, independent of the path before $\tau_a$ . This allows computation of hitting probabilities for multiple levels.

Applic ation: Gambler's ruin

ExampleHitting two levels

What is $\mathbb{P}(B_\tau = b \mid B_0 = x)$ , where $\tau = \inf\{t : B_t \in \{a, b\}\}$ with $a < x < b$ ? By the strong Markov property and martingale arguments, the probability is $(x-a)/(b-a)$ .

Summary

The strong Markov property is fundamental to analyzing Brownian motion at random times, with applications to hitting times, exit problems, and optimal stopping.