Euler–Rodrigues formula

2014-01-28T13:05:34Z

132.229.212.81: /* Connection with SU(2) spin matrices */

{{context|date=August 2011}}
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]]

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Euler–Rodrigues formula