Basic Affine Jump Diffusion

In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form

:$dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt\,dB_t+dJ_t,\qquad t\geq 0, Z_\geq 0,$

where $B$ is a standard Brownian motion, and $J$ is an independent compound Poisson process with constant jump intensity $l$ and independent exponentially distributed jumps with mean $\mu$. For the process to be well defined, it is necessary that $\kappa \theta \geq 0$ and $\mu \geq 0$. 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, since both the moment generating function

:$m\left( q\right) =\operatorname \left( e^\right) ,\qquad q\in \mathbb,$

and the characteristic function

:$\varphi \left( u\right) =\operatorname \left( e^\right) ,\qquad u\in \mathbb,$

are known in closed form.

The characteristic function allows one to calculate the density of an integrated basic AJD

:$\int_0^t Z_s \, ds$

by Fourier inversion, which can be done efficiently using the FFT.

References

{{Reflist, refs=

{{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 , doi=10.2469/faj.v57.n1.2418, s2cid = 12334040 }
Preprint
/ref>
{{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 , issue = 4 , pages = 8–26 , doi=10.3905/jod.2006.635417}
Preprint
/ref>
{{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 , doi=10.21314/JCF.2009.200}
Preprint
/ref>
{{cite journal , author = Peter Feldhütter, Mads Stenbo Nielsen , year = 2010 , title = Systematic and idiosyncratic default risk in synthetic credit markets}
Preprint
/ref>

Stochastic processes