Basic affine jump diffusion explained

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,} 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 ^{t} _{0} Z _{s} ds} \right) , q\in R ,

and the characteristic function

\varphi \left( u\right) = \operatorname{E} \left( e ^{iu\int ^{t} _{0} Z _{s} ds} \right) , u\in R ,

are known in closed form.

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

\int ^{t} _{0} Z _{s} ds

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