Brownian Motion as a Markov Process

Markov property

Brownian motion is a Markov process: given $B_t$, the future distribution of $(B_r)_{r>t}$ does not depend on earlier values beyond time $t$.

Why

For $r>t$,

$$ B_r=B_t+(B_r-B_t). $$

The increment $B_r-B_t$ is independent of $\mathcal F_t$, so once $B_t$ is known, the future only depends on fresh independent Gaussian increments.

Transition law

Conditionally on $B_t=x$,

$$ B_r\mid B_t=x \sim N(x,r-t). $$

Source Links

• STA447
• Phase 6 Atomic Reading Order
• Phase 6 — Brownian Motion 基础
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