Basic affine jump diffusion

"AJD" redirects here. For other uses, see AJD (disambiguation).

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

$d{Z}_t=\kappa (\theta -Z_t)dt+\sigma \sqrt{Z_t}d{B}_t+d{J}_t,t\ge 0,Z_0\ge 0,$

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 $\kappa \theta \ge 0$ and $\mu \ge 0$. 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

$m\left(q\right)=\mathrm{E}\left(e^{q\underset{0}{\overset{t}{\int }}{Z}_{s}ds}\right),q\in ℝ,$

and the characteristic function

$\phi \left(u\right)=\mathrm{E}\left(e^{iu\underset{0}{\overset{t}{\int }}{Z}_{s}ds}\right),u\in ℝ,$

are known in closed form.

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

$\underset{0}{\overset{t}{\int }}{Z}_{s}ds$

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