## Basic affine jump diffusion

In probability theory, a basic affine jump diffusion (basic AJD) is a 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 μ. 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 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

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

and the characteristic function

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

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.