Basic affine jump diffusion

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

${\displaystyle dZ_{t}=\kappa (\theta -Z_{t})\,dt+\sigma {\sqrt {Z_{t}}}\,dB_{t}+dJ_{t},\qquad t\geq 0,Z_{0}\geq 0,}$

where ${\displaystyle B}$ is a standard Brownian motion, and ${\displaystyle J}$ is an independent compound Poisson process with constant jump intensity ${\displaystyle l}$ and independent exponentially distributed jumps with mean ${\displaystyle \mu }$. For the process to be well defined, it is necessary that ${\displaystyle \kappa \theta \geq 0}$ and ${\displaystyle \mu \geq 0}$. 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,^[1][2][3][4] since both the moment generating function

${\displaystyle m\left(q\right)=\operatorname {E} \left(e^{q\int _{0}^{t}Z_{s}\,ds}\right),\qquad q\in \mathbb {R} ,}$

and the characteristic function

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

are known in closed form.^[3]

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

${\displaystyle \int _{0}^{t}Z_{s}\,ds}$

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