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.
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.
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.