Basic affine jump diffusion

Short description: Stochastic process

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

[math]\displaystyle{ dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt{Z_t}\,dB_t+dJ_t,\qquad t\geq 0, Z_{0}\geq 0, }[/math]

where [math]\displaystyle{ B }[/math] is a standard Brownian motion, and [math]\displaystyle{ J }[/math] is an independent compound Poisson process with constant jump intensity [math]\displaystyle{ l }[/math] and independent exponentially distributed jumps with mean [math]\displaystyle{ \mu }[/math]. For the process to be well defined, it is necessary that [math]\displaystyle{ \kappa \theta \geq 0 }[/math] and [math]\displaystyle{ \mu \geq 0 }[/math]. 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

[math]\displaystyle{ m\left( q\right) =\operatorname{E} \left( e^{q\int_0^t Z_s \, ds}\right) ,\qquad q\in \mathbb{R}, }[/math]

and the characteristic function

[math]\displaystyle{ \varphi \left( u\right) =\operatorname{E} \left( e^{iu\int_0^t Z_s \, ds}\right) ,\qquad u\in \mathbb{R}, }[/math]

are known in closed form.

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

[math]\displaystyle{ \int_0^t Z_s \, ds }[/math]

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