How Do You Integrate Complex Functions with Fractional Powers?

[cernlife]

I'm struggling to work out how to integrate the following

\int_0^t(\gamma^{1/\kappa}-i\zeta{w}(1-t/s)_+^{H-1/2})^{\kappa}ds

here (.)_+ denotes the positive part

if I did not have the ^(H-1/2) I can do it, alas it does have it! and so it stumps me on how to evaluate this integral.

any advice much appreciated

 

[cernlife]

sorry, it was meant to be (1-s/t)_+ that was a typo...

although I did enter it correctly into Mathematica...

basically what I am looking at is a theorem from the paper "fractional tempered stable motion" and also work from the paper "Integrating volatility clustering into exponential Levy models"which states that a convoluted subordinator is defined as

X_t = \int_0^t G(t,s)dL(s)

which a theorem then states that if L(s) is a tempered stable Levy process the X_t is also tempered stable. where G(t.s) is some kernal of volterra type.

basically, the characteristic function of X_t can then be computed by

E[e^{i\zeta{X_t}}]=e^{\int_0^t \psi(\zeta G(t,s))ds}

where \psi(\zeta) is the cumulant generating function, which for the tempered stable is defined as

\psi(\zeta)=\gamma\delta-\delta(\gamma^{1/\kappa}-2i\zeta)^{\kappa}

now chosing the kernal to be adamped version of the fractional Holmgren-Liouville integral, i.e

G_H(t,s)=\frac{H+1/2}{E[L(1)]}\left(1-\frac{s}{t}\right)_+^{H-1/2}

I am then left with trying to work out the following integral

\int_0^t \gamma\delta-\delta(\gamma^{1/\kappa}-2i(\zeta\frac{H+1/2}{E[L(1)]}\left(1-\frac{s}{t}\right)_+^{H-1/2}))^{\kappa}ds

let w=\frac{H+1/2}{E[L(1)]} and pull out what we can from the front of the integral, I then have

\gamma\delta{t}-\delta\int_0^t (\gamma^{1/\kappa}-2i\zeta{w}\left(1-\frac{s}{t}\right)_+^{H-1/2})^{\kappa}ds

which is what I need to integrate, and is where I am not sure at all where to start...

any reply's, much appreciated.

 

[henry_m]

By 'the positive part of f' do you mean 0 when f is negative and f when f is positive? In which case, you can drop the '+' entirely, and the put x=1-s/t. What you're left with will be something like
\int_0^1 \mathrm{d}x(1-iAx^\alpha)^\beta
for which I don't think there'll be a nice closed form solution in general. Wolfram Alpha gives the answer as a hypergeometric function.