Characteristic Function of a Compound Poisson Process

[mikhairu]

Hello,

I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(.

I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed.

So, in trying to find the CF of X I do the following:
(Notation: CFy = Characteristic Function of y (y is subscript))

CF(X) = E[exp{i*u*X}]
= E[ (E[exp{i*u*Y}])^N ]
= E[ (CFy(u))^N ]
= E[ (exp{ ln[CFy(u)] })^N ]
which is really just a moment generating function Phi_N (Phi subscript N):
Phi_N( ln[CFy(u)] ).

I don't know how to go from here... Y's are Normally distributed but the entire process is Poisson.. so I'm not sure how to combine these. Please help!

Thank you.

 

[bpet]

Hello,

I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(.

I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed.

So, in trying to find the CF of X I do the following:
(Notation: CFy = Characteristic Function of y (y is subscript))

CF(X) = E[exp{i*u*X}]
= E[ (E[exp{i*u*Y}])^N ]
= E[ (CFy(u))^N ]
= E[ (exp{ ln[CFy(u)] })^N ]
which is really just a moment generating function Phi_N (Phi subscript N):
Phi_N( ln[CFy(u)] ).

I don't know how to go from here... Y's are Normally distributed but the entire process is Poisson.. so I'm not sure how to combine these. Please help!

Thank you.

After line 3 it should be possible to evaluate expressions of the form E[z^N] using P[N=n]=exp(-L*t)*(L*t)^n/n! where L is the rate of the Poisson process.

 

[mikhairu]

Thank you for your reply!

The full answer is here: http://math.stackexchange.com/questions/31329/distribution-of-compound-poisson-process