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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