title: Basic affine jump diffusion
text: # Basic affine jump diffusion
  
  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.
  
  