Just a little confused about the following. Given that denotes a one-dimensional standard Brownian motion, find

Step 1: Assume that , then

Step 2: Then, in general:

Could someone explain each step and properties used? Thanks.

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- March 9th 2009, 09:33 PMhoran[SOLVED] Covariance of Brownian Motion
Just a little confused about the following. Given that denotes a one-dimensional standard Brownian motion, find

Step 1: Assume that , then

Step 2: Then, in general:

Could someone explain each step and properties used? Thanks. - March 9th 2009, 10:18 PMzigzag20
Fix 0 ≤ s ≤ t.

Step 1:

Cov(W(t), W(s)) = E[W(t)W(s)] − E[W(t)]E[W(s)] = E[W(t) * W(s)] as these are standard brownian motion. Hence, E(W(t)) = E(W(s)) = 0.

E[W(t)W(s)] = E[(W(s) + W(t) − W(s))W(s)]

= E[W^2(s)] + E[(W(t) − W(s))W(s)] = s

Why? Since E[W(t)] = 0 for all t and since by definition of the standard Brownian motion we have E[W^2(s)] = s.

And by independent increments property we have E[(W(t) − W(s))W(s)] = E[(W(t) − W(s))(W(s) − W(0))] = E[(W(t) − W(s)]E[(W(s) − W(0)] = 0, since

increments have a zero mean Gaussian distribution.

Step 2: Repeat the same for t < s. Hence, you shall see min(t,s) to be the general answer.

- March 23rd 2009, 07:45 AMsonyrobocup
I am reading the solution, and wondering in W^2(s), what does the ^ symbol stands for.