Calculate Fourier transform for the characteristic function of a rv

[mihai.rd]

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

 

[mihai.rd]

Bump!
103 views no hints?

 

[Mute]

It's not really clear what exactly you're trying to do, or what you've done so far. Explicitly showing us some of your calculation may help.

What exactly are you seeking as a solution? The characteristic function of z in terms of the characteristic function of x?

I have to admit, it's not clear to me where your expressions are coming from. My first instinct would be to simply calculate $\phi_Z(u)$ as

$$\phi_Z(u) = \mathbb{E}\left[e^{iuZ}\right] = \mathbb{E}\left[e^{iu~\rm{max}(X,0)}\right];$$
assuming X has a probability density, you could easily write this down in terms of integrals over the density. Since this is for any continuous random variable x, that would be a legitimate answer, unless you have some specific form you have in mind, or if you need a more general form for distributions of X which don't have well-defined probability densities?

 

[mihai.rd]

Indeed this is what my final goal is: calculate the characteristic function of Z in terms of the characteristic function of X.
The problem is that in my case X has a more complicated distribution, namely being the solution of stochastic differential equation for a double exponential jump difussion with stochastic volatility underlying asset.

The only straighforward (and calculable result) is by applying the above mentioned formula involving the Fourier transform.

 

[Mute]

I worked through it a little bit myself, and I get an expression similar to what you have written down. That is, you already have an expression for the characteristic function of Z in terms of the characteristic function of X (and some integrals of it).

Are you actually trying to compute the probability density of Z, which is the inverse Fourier transform of the characteristic function for Z?

 

[mihai.rd]

Actually, what i am trying to check is the result for the expression of the characteristic function of the Z variable in terms in terms of the X variable, that is using the inverse Fourier transform.
If i implement the formula in Matlab using FFT, then the results don't match (direct formula vs inverse Fourier for the standard normal); and i have tried matching the results also via quadrature integration and improper integration.

Could you please give some details on how you worked out the inverse Fourier formula and if you can get the same results for the standard normal say for three points (e.g. u=1, 2, 3).

I would really appreciate it as it is my final problem in a master thesis.

Many thanks,
Mihai

 

[mihai.rd]

Still in the clouds with this one.
Considering offering incentives :)