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Hi all,
does somebody know if there is an analytic formula for the expectation of a CEV variable to a power?

I.e.   dF = v * F^b

where v,b are constants. I am looking for 

E[F^a] 

for some real constant a. (we know the density of F but I'd rather not try to do the integral if this result is known...)

Thanks!

I suspect the answer is "yes, but only if 'a' and 'b' have certain relationships".
An approach is to use Ito to change the CEV sde to dX_t = A dt + B sqrt(X) dW by taking X to be a power of F (see below).
Then X_t  should have an easy characteristic function phi(t,x; z) = E[e^(i z X_t | X_0 = x], as the pde generator is affine.

Then, you can differentiate phi to get analytic formulas for moments E[ (X_t)^n] = E[ (F_t)^(2-2b)n], if I haven't made a mistake.
So then the relationships I mention must be: a = (2 - 2b) n, for n = 1,2,....
For other 'a', you may be stuck with using the transition density for F.  

Thanks for that. My immediate interest is actually for a=b, so I am probably stuck trying to do the integral (or letting mathematica have a crack). I was hoping somebody may know a reference where this is written down.

Though Alan has already suggested a good answer, I have tried to answer your question in terms of my recent research but you will have to become familiar with some other basic concepts like Z-series of stochastic processes etc.

I hope you find the post in link below helpful.
https://forum.wilmott.com/viewtopic.php?f=4&t=99702&p=870260#p870260

Thanks for that. My immediate interest is actually for a=b, so I am probably stuck trying to do the integral (or letting mathematica have a crack). I was hoping somebody may know a reference where this is written down.

I just looked at the density (eqn (1) in Schroder).
With that, it looks like your general case is indeed expressible in terms of the confluent hypergeometric function, 
as Mathematica can do the general integral:

Integrate[w^p E^(-w) BesselI[k,2 Sqrt[x w]],{w,0,Infinity}]
=  x^(k/2) Gamma[1+k/2+p] Hypergeometric1F1Regularized[1+k/2+p,1+k,x]

subject to some conditions on the parameters: p > -1 && (k/2)+p > -1. 
Also Hypergeometric1F1Regularized(a,b,z)  = [$] M(a,b,z)/\Gamma(b)[$] in more standard notation. 

So, clean it up and you've got your answer.

I attach the relevant pages from Schroder: And in normal notation we have, subject to the conditions above,

[$] \int_0^{\infty} w^p  \, e^{-w} I_k ( 2 \sqrt{x w} ) \, dw = \frac{\Gamma(1 + p + k/2)}{\Gamma(1+k)} x^{k/2} M(1 + p + k/2, 1 + k, x)[$]

Excellent, thank you very much! Was hoping it would be a simpler expression but still useful to have a closed form.