Basic affine jump diffusion

In 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.

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