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.