Английская Википедия:Basic affine jump diffusion

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Шаблон:Short description Шаблон:Technical Шаблон:Redirect In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form

<math> 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> B </math> is a standard Brownian motion, and <math> J </math> is an independent compound Poisson process with constant jump intensity <math> l </math> and independent exponentially distributed jumps with mean <math> \mu </math>. For the process to be well defined, it is necessary that <math> \kappa \theta \geq 0 </math> and <math> \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> 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> \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> \int_0^t Z_s \, ds </math>

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