6 Brownian motion as a Markov process

We have seen in 2.13 that for a d-dimensional Brownian motion (Bt)t≥0 and any s > 0 the shifted process Wt := Bt+s - Bs, t ≥ 0, is again a BMd which is independent of (Bt)0≤t≤s. Since Bt+s = Wt + Bs, we can interpret this as a renewal property: Rather than going from 0 = B0 straight away to x = Bt+s in (t + s) units of time, we stop after time s at x′ = Bs and move, using a further Brownian motion Wt for t units of time, from x’ to x = Bt+s. This situation is shown in Figure 6.1:


Fig. 6.1. Wt := Bt+s - Bs, t ≥ 0, is a Brownian motion in the new coordinate system with origin (s, Bs).

6.1 The Markov property ...

Get Brownian Motion, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.