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
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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 http://www-m4.ma.tum.de/Papers/Rosinski/tstable.pd or http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf) 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
\(\displaystyle M(A)=\int_{R^d}\int_0^{\infty} I_A(tx)t^{-\alpha-1}e^{-t}dtR(dx)\)
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So the question is, how can I work out what \(\displaystyle Q\) is? and what is \(\displaystyle R(dx)\)?
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I would greatly appreciate if anyone can help me on this one.
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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 http://www-m4.ma.tum.de/Papers/Rosinski/tstable.pd or http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf) 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
\(\displaystyle M(A)=\int_{R^d}\int_0^{\infty} I_A(tx)t^{-\alpha-1}e^{-t}dtR(dx)\)
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So the question is, how can I work out what \(\displaystyle Q\) is? and what is \(\displaystyle R(dx)\)?
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I would greatly appreciate if anyone can help me on this one.