Brownian motion

Question

Verify that $E( X(t) X(s) | X(0)=0 ) = min (t, s)$, where $X(t)$ is standard Brownian motion. I don't know where to start. Thanks!

What is the definition of a Brownian motion in your textbook? (My question is motivated by your reaction to Nate's hint.)
@Didier: My book is very difficult, and we don't use it too much.

Nate Eldredge · Accepted Answer · 2011-04-25 03:43:02Z

up vote 3 down vote accepted

Hint: if $t > s$, write $X(t) = X(s) + (X(t) - X(s))$.

answered Apr 25 '11 at 3:43

Nate Eldredge
22.4k12563

	@Nate: Thanks. Can you show a few more steps? – user9636 Apr 25 '11 at 4:02
	@user9636: If $t>s$ then $X(s)$ and $X(t)-X(s)$ are independent standard Brownian motions. This may be useful when you take the expectation of their product. – Henry Apr 25 '11 at 16:33
	@Henry Sorry but $X(s)$ and $X(t)-X(s)$ are not Brownian motions. They are independent centered Gaussian random variables (with respective variance $s$ and $t-s$). – Did Apr 25 '11 at 19:56
	@Didier: True enough, I was careless when abusing the language in the question. I was concentrating on independent and zero mean. – Henry Apr 25 '11 at 21:13
	@Nate: Thanks for your hint. Finally, it was useful. I got it. – user9636 Apr 28 '11 at 15:06

1 Answer