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

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

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.