Brownian Motion Question HELP

[RobD]

Hey I am in a stochastic calculus class for finance I have a question on a problem that arised in my class I am really lost and have no idea how to treat the brownian motion with a function in it. I couldnt seem to find anything in Shreve that really helped.

Let \(W(t) t>= 0\) be a standard brownian motion

Calculate \(E[Y(t)Y(s)]\) where \(Y(t)= (e^{-t} )(W(e^{2t}))\)

Any help or direcion would be greatly appreciation

 

[AleNinho]

From the theory you have

\(E[W(t)W(s)]=t \land s\)

substituting \(s \rightarrow g(s)\)

\(E[W(g(t))W(g(s))]=g(t) \land g(s)\)

Then it's easy to see that

\(E[Y(s) Y(t)]=e^{-|t-s|}\)

 

[RobD]

Ok grreat that's what I figured but I couldn't find anything to back me up. Thank you