mihai.rd
- 5
- 0
Homework Statement
In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that:
F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv)
where F_{l,v}(g(l)) is the Fourier transform of g(l) and
g(l)=E[e^{iuX}|X>l]−Prob(X>l)
Then the characteristic function for Z follows by:
\phi_{Z}(u) =E [ e^{iuz} ] = F^{-1}_{0,v}[ \phi_{X}(u+v) - \phi_{X}(u) ] / (iv) +1
The attempt at a solution
The first approach was to calculate straigthforward both the Fourier transform and the inverse , but i can't get around the double integral.
Any suggestions are highly appreciated.
Many thanks,
Mihai
In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that:
F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv)
where F_{l,v}(g(l)) is the Fourier transform of g(l) and
g(l)=E[e^{iuX}|X>l]−Prob(X>l)
Then the characteristic function for Z follows by:
\phi_{Z}(u) =E [ e^{iuz} ] = F^{-1}_{0,v}[ \phi_{X}(u+v) - \phi_{X}(u) ] / (iv) +1
The attempt at a solution
The first approach was to calculate straigthforward both the Fourier transform and the inverse , but i can't get around the double integral.
Any suggestions are highly appreciated.
Many thanks,
Mihai