Basic affine jump diffusion

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

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

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

: $m\left( q\right) =\operatorname{E} \left( e^{q\int_0^t Z_s \, ds}\right)
,\qquad q\in \mathbb{R},$

and the characteristic function

: $\varphi \left( u\right) =\operatorname{E} \left( e^{iu\int_0^t Z_s \, ds}\right) ,\qquad u\in \mathbb{R},$

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 }} [http://www.darrellduffie.com/uploads/working/DuffieGarleanu2000.pdf Preprint]

{{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}} [http://w4.stern.nyu.edu/salomon/docs/Credit2006/AM_BasketIntensities_wp05.pdf Preprint]

{{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}} [http://www.eckner.com/papers/bAJD_comp.pdf Preprint]

{{cite journal

| last1 = Feldhutter | first1 = P.

| last2 = Nielsen | first2 = M. S.

| date = January 2012

| doi = 10.1093/jjfinec/nbr011

| issue = 2

| journal = Journal of Financial Econometrics

| pages = 292–324

| title = Systematic and idiosyncratic default risk in synthetic credit markets

| url = http://www.feldhutter.com/CDOpaper070710.pdf

| volume = 10}}

}}

Category:Stochastic processes