complex integral to evaluate

cernlife

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

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

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

Fremen

Is that related to fractional Brownian motion?

Tried firing it into Wolfram alpha?

cernlife

I tried using Mathematica (made by Wolfram) and it couldn't do it, (it could when the (H-1/2) was not in it) but otherwise could not integrate it

Fremen

I think you need to give a bit more information about what we're looking at here. What values could H take? If you're integrating from 0 to t, then s<t,
so [latex](1-t/s)_+[/latex] is always 0. Are you sure you've written the integral correctly?

If you break the integral up into two parts, one where the max function is 0 and one where it's nonzero, then maybe mathematica will fare better.

I've never come across an integral that I could do but that mathematica couldn't, so if it's no help you may have to integrate numerically.

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

[latex]X_t = \int_0^t G(t,s)dL(s)[/latex]

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

basically, the characteristic function of [latex]X_t[/latex] can then be computed by

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

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

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

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

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

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

[latex]\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[/latex]

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

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

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

Thank you for your previous reply's, much appreciated.

Fremen

Again, there's no need for the positive part symbol since that term is always positive. The only other thing I could recommend is the "Assumptions" keyword in mathematica.

Mathematica always assumes the most general form for variables - this gets it into trouble sometimes. Try Assumptions -> H>0 && gamma>0 and whatever else you already know.