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.