Euler–Rodrigues formula: Difference between revisions
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In [[probability theory]], a '''basic affine jump diffusion (basic AJD)''' is a [[stochastic process]] Z of the form | |||
:<math> dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt{Z_t}\,dB_t+dJ_t,\qquad t\geq 0, | |||
Z_{0}\geq 0, </math> | |||
where <math> B </math> is a standard [[Brownian motion]], and <math> J </math> is an independent [[compound Poisson process]] with constant jump intensity <math> l </math> and independent exponentially distributed jumps with mean <math> \mu </math>. For the process to be well defined, it is necessary that <math> \kappa \theta \geq 0 </math> and <math> \mu \geq 0 </math>. A basic AJD is a special case of an [[affine process]] and of a [[jump diffusion]]. On the other hand, the [[Cox–Ingersoll–Ross]] (CIR) process is a special case of a basic AJD. | |||
Basic AJDs are attractive for modeling default times in [[credit risk]] applications,<ref name="DufGar01"/><ref name="Mor06"/><ref name="Eck09" /><ref name="FelNie10"/> since both the [[moment generating function]] | |||
:<math> m\left( q\right) =\operatorname{E} \left( e^{q\int_0^t Z_s \, ds}\right) | |||
,\qquad q\in \mathbb{R}, </math> | |||
and the [[characteristic function (probability theory)|characteristic function]] | |||
:<math> \varphi \left( u\right) =\operatorname{E} \left( e^{iu\int_0^t Z_s \, ds}\right) ,\qquad u\in \mathbb{R}, </math> | |||
are known in [[Closed-form expression|closed form]].<ref name="Eck09">{{cite journal | author = Andreas Ecker | year = 2009 | title = Computational Techniques for basic Affine Models of Portfolio Credit Risk | journal = Journal of Computational Finance | volume = 13 | pages = 63–97}}</ref> | |||
The characteristic function allows one to calculate the density of an integrated basic AJD | |||
:<math> \int_0^t Z_s \, ds </math> | |||
by [[Fourier inversion]], which can be done efficiently using the [[Fast Fourier transform|FFT]]. | |||
==References== | |||
{{Reflist|refs= | |||
<ref name="DufGar01">{{cite journal | author = Darrell Duffie, Nicolae Gârleanu | year = 2001 | title = Risk and Valuation of Collateralized Debt Obligations | journal = Financial Analysts Journal | volume = 57 | pages = 41–59}} [http://www.darrellduffie.com/uploads/working/DuffieGarleanu2000.pdf Preprint]</ref> | |||
<ref name="Mor06">{{cite journal | author = Allan Mortensen | year = 2006 | title = Semi-Analytical Valuation of Basket Credit Derivatives in Intensity-Based Models | journal = Journal of Derivatives | volume = 13 | pages = 8–26}} [http://w4.stern.nyu.edu/salomon/docs/Credit2006/AM_BasketIntensities_wp05.pdf Preprint]</ref> | |||
<ref name="Eck09">{{cite journal | author = Andreas Ecker | year = 2009 | title = Computational Techniques for basic Affine Models of Portfolio Credit Risk | journal = Journal of Computational Finance | volume = 13 | pages = 63–97}} [http://www.eckner.com/papers/bAJD_comp.pdf Preprint]</ref> | |||
<ref name="FelNie10">{{cite journal | author = Peter Feldhütter, Mads Stenbo Nielsen | year = 2010 | title = Systematic and idiosyncratic default risk in synthetic credit markets}} [http://www.feldhutter.com/CDOpaper070710.pdf Preprint]</ref> | |||
}} | |||
[[Category:Stochastic processes]] | |||
{{probability-stub}} | |||
Revision as of 14:05, 28 January 2014
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In probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form
where is a standard Brownian motion, and is an independent compound Poisson process with constant jump intensity and independent exponentially distributed jumps with mean . For the process to be well defined, it is necessary that and . A basic AJD is a special case of an affine process and of a jump diffusion. On the other hand, the Cox–Ingersoll–Ross (CIR) process is a special case of a basic AJD.
Basic AJDs are attractive for modeling default times in credit risk applications,[1][2][3][4] since both the moment generating function
and the characteristic function
are known in closed form.[3]
The characteristic function allows one to calculate the density of an integrated basic AJD
by Fourier inversion, which can be done efficiently using the FFT.
References
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