Tempered Stable Distributions - Computing R(dx)

cernlife

Im trying to compute R(dx) from a paper by Jan Rosinski, which can be found here, (also here using slightly different notation).

In the paper on page 3 we have the following theorem

Theorem 2.3. The Levy measure M of a tempered alpha stable distribution can be written in the form

[latex]M(A) = \int_{R^d}\int_0^{\infty} I_A(tx)t^{-\alpha-1}e^{-t}dtR(dx)[/latex]

where I_A(tx) I assume is the indicator funtion, i.e. tx is defined on the interval A

now I'm using definition of the Gamma function kernal to say

[latex]\int_0^{\infty} t^{-\alpha-1}e^{-t}dt = \Gamma(-\alpha)[/latex]


I know the Levy measure M, and so putting that in for M, I then had

[latex]2^{\alpha}\delta\frac{\alpha}{\Gamma(1-\alpha)}x^{-1-\alpha}e^{-0.5\gamma^{1/\alpha}x} = \Gamma(-\alpha)\int_{R^d}R(dx)[/latex]

So I need to work out what the function R(dx) is, which is my problem. I was thinking I could just differentiate both sides which would get the integral out of the right hand side and then I could easily rearrange to find R(dx). However I am not sure about this method as I have this integral over the range R^d.

How do I work out what R(dx) is?

I was told in another forum by someone...

"From the statement of the theorem, it seems that equation (2.6) holds for a given Borel measure satisfying the finiteness criterion (2.8). Is this characterization of the measure not enough for your computations?"

However I fail to see how this helps. Please any help in how I compute R(dx) would be much appreciated.

Fremen

First observation is that you dropped your indicator function going from line 1 to line 3...

cernlife

Thanks for the reply, a little more info on my problem is need I think.

A tempered stable distribution is when a stable distribution is tempered by an exponential function of the form [latex]e^{-\theta{x}}[/latex]. In my particular case we are using a tempered stable law defined by Barndorff-Nielsen in the paper "modified stable processes" found here, http://economics.ouls.ox.ac.uk/12286/1/nmsprocnew1.pdf.

In Barndorff's paper, [latex]\theta = (1/2)\gamma^{1/\alpha}[/latex], hence the tempering function is defined as [latex]e^{-(1/2)\gamma^{1/\alpha}{x}[/latex]. In Rosinski's paper he states that tempering of the stable density [latex]f \mapsto f_{\theta}[/latex] leads to tempering of the corresponding Levy measure [latex]M \mapsto M_{\theta}[/latex], where [latex]M_{\theta}(dx) = e^{-\theta{x}}M(dx)[/latex].

Rosinski then goes on to say the Levy measure of a stable law in polar coordinates is of the form

[latex]M_0(dr, du) = r^{-\alpha-1}dr\sigma(du)[/latex] (2.1)

and then says the Levy measure of a tempered stable density can be written as

[latex]M(dr, du) = r^{-\alpha-1}q(r,u)dr\sigma(du)[/latex] (2.2)

he then says, the tempering function q in (2.2) can be represented as

[latex]q(r,u) = \int_0^{\infty}e^{-rs}Q(ds|u)[/latex] (2.3)

now I'm assuming in my particular case, [latex]q(r,u) = e^{-(1/2)\gamma^{1/\alpha}{r}[/latex] although by now I'm getting lost and unsure of what Im saying.

Rosinski's paper also defines a measure R by

[latex]R(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}Q(dx)[/latex] (2.5)

and has

[latex]Q(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}R(dx)[/latex] (2.6)

now I know that for my particular tempered stable density the levy measure M is given by

[latex]2^{\alpha}\delta\frac{\alpha}{ \Gamma(1-\alpha)}x^{-1-\alpha}e^{-(1/2)\gamma^{1/\alpha}x}dx[/latex]

So im even more confused than ever now, how can I work out what [latex]Q[/latex] is? what is [latex]a[/latex]? how do I use this to work out what R(dx) is? I feel that Im just totally missing the point!

I really fails to understand (2.5) and (2.6) as I fail to understand what [latex]||x||^{\alpha}[/latex] means, moreover i dont understand what [latex]I_A(x/||x||^2)[/latex] means.

Fremen

I'm not really too comfortable with seeing a levy measure in polar coordinates - I'm not used to it and it's throwing my intuition off completely.

Have you had a look at Cont and Tankov's book on Levy processes? They have a section on exponential tilting. It may be some help.