I don't understand this one small point about Brownian motion...
Here's the question, my answer and the part I always miss out...
Let be a standard Brownian motion. Show that.
(a) is a standard Brownian motion, where
Question is easy enough. Can show that almost surely and get that for , is distributed with mean and variance and is independent of . (Not ENTIRELY sure why this is though)
Then comes this small statement that I don't know how to show (and hence don't know if it's true).
has continuous paths.
How do I show that? It's only ever stated in answers so I don't need some in depth proof, just a simple, it has continuous paths because...
Now if you consider any f that is bounded and continuous then is bounded and continuous hence
A rather long and boring way of proving what your intuition tells you.