Here is a converse to roughly speaking, if all the projections of a d-dimensional process X onto hyper planes are one-dimensional Brownian motions, then X is a d-dimensional Brownian motion. Suppose...

Here is a converse to roughly speaking, if all the projections of a d-dimensional process X onto hyper planes are one-dimensional Brownian motions, then X is a d-dimensional Brownian motion.

Suppose
  is a d-dimensional continuous process, i.e., one taking values in

be the minimal augmented filtration generated by X . Suppose that whenever

with
  is a one-dimensional Brownian motion started at 0 with respect to the filtration

(1) If
  Calculate

(3) Prove that
  is a d-dimensional Brownian motion started from 0. (Some care is needed with the filtrations. If we only know that
  is a Brownian motion with respect to the filtration generated by Y^λ
for each
  the assertion is not true.

Chapter 3