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Thread: compute R(dx).

  1. #1
    May 2011

    compute R(dx).

    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, $\displaystyle \theta = (1/2)\gamma^{1/\alpha}$, hence the tempering function is defined as $\displaystyle e^{-(1/2} \gamma^{1/\alpha}{x}$.

    In Rosinski's paper on "tempering stable processes" (which can be found or he states that tempering of the stable density $\displaystyle f \mapsto f_{\theta}$ leads to tempering of the corresponding Levy measure $\displaystyle M \mapsto M_{\theta}$, where $\displaystyle M_{\theta}(dx) = e^{-\theta{x}}M(dx)$.

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

    $\displaystyle M_0(dr, du) = r^{-\alpha-1}dr\sigma(du) \hspace{30mm} (2.1)$

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

    $\displaystyle M(dr, du) = r^{-\alpha-1}q(r,u)dr\sigma(du) \hspace{30mm} (2.2)$

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

    $\displaystyle q(r,u) = \int_0^{\infty}e^{-rs}Q(ds|u) \hspace{30mm} (2.3)$

    Rosinski's paper also defines a measure R by

    $\displaystyle R(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}Q(dx) \hspace{30mm} (2.5)$

    and has

    $\displaystyle Q(A) = \int_{R^d} I_A(x/||x||^2)||x||^{\alpha}R(dx) \hspace{30mm} (2.6)$

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

    $\displaystyle 2^{\alpha}\delta\frac{\alpha}{ \Gamma(1-\alpha)}x^{-1-\alpha}e^{-(1/2)\gamma^{1/\alpha}x}dx$

    Rosinski then goes on to state Theorem 2.3: The Levy measure $\displaystyle M$ of a tempered stable distribution can be written in the form

    $\displaystyle M(A)=\int_{R^d}\int_0^{\infty} I_A(tx)t^{-\alpha-1}e^{-t}dtR(dx)$


    So the question is, how can I work out what $\displaystyle Q$ is? and what is $\displaystyle R(dx)$?


    I would greatly appreciate if anyone can help me on this one.
    Last edited by mr fantastic; May 28th 2011 at 02:51 PM. Reason: Title.
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