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

My question is how to compute R(dx). But before I can ask that I have to write down the background to my problem, so bear withme ---------------------------------------------------------------------------- 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.oul...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 on "tempering stable processes" (which can be found here or here) 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) \hspace{30mm} (2.1)[/latex] 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) \hspace{30mm} (2.2)[/latex] 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) \hspace{30mm} (2.3)[/latex] Rosinski's paper also defines a measure R by [latex]R(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}Q(dx) \hspace{30mm} (2.5)[/latex] and has [latex]Q(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}R(dx) \hspace{30mm} (2.6)[/latex] 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] Rosinski then goes on to state Theorem 2.3: The Levy measure M of a tempered 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] ---------------------------------------------------------------------- So the question is, how can I work out what [latex]Q[/latex] is? and what is [latex]R(dx)[/latex]?