AJD

In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form d Z t = κ ( θ − Z t ) d t + σ Z t d B t + d J t , t ≥ 0 , Z 0 ≥ 0 , {\displaystyle 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 {\displaystyle B} is a standard Brownian motion, and J {\displaystyle J} is an independent compound Poisson process with constant jump intensity l {\displaystyle l} and independent exponentially distributed jumps with mean μ {\displaystyle \mu } . For the process to be well defined, it is necessary that κ θ ≥ 0 {\displaystyle \kappa \theta \geq 0} and μ ≥ 0 {\displaystyle \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 ( q ) = E ⁡ ( e q ∫ 0 t Z s d s ) , q ∈ R , {\displaystyle 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 φ ( u ) = E ⁡ ( e i u ∫ 0 t Z s d s ) , u ∈ R , {\displaystyle \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 ∫ 0 t Z s d s {\displaystyle \int _{0}^{t}Z_{s}\,ds} by Fourier inversion, which can be done efficiently using the FFT.

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