2009年3月13日 星期五

Formal solution to the SDEs

Recall the SDEs in part one of the paper by Yariv are (in differential form):
\begin{align*}d\delta&=-\omega_1\delta dt+dZ_i\\d\psi&=(A\omega_1-B)\delta dt-AdZ_i-dZ_r\end{align*} 
where Zi and Zr are the driving Levy noise process.

The soultion to the amplitude flucturation is simply the OU-process:
\delta(t)=\delta(0)e^{-\omega_1t}+\int_0^te^{-\omega_1(t-\tau)}dZ_i(\tau)

The solution to the phase flucturation can be derived by using integration by part:
\psi(t)=\psi(0)+(\alpha-A)(\delta(t)-\delta(0))-(\alpha Z_i(t)+Z_r(t)) 
where alpha is defined in (25).

We can find the joint ch.f of the amplitude and the phase driven by an arbitary 2-dimensional Levy noise process; just follow the trick in the proof of Lemma 15.1 in Peter Tankov’s book. However, this expression is too general to be useful for the further analysis.

To make life easer, we make the following assumptions:

  1. Both of Zi and Zr are symmetric alpha-stable processes (SaS for short)
  2. The driving noise Zi and Zr are independent

The assumption one makes it possible to express the ch.f in closed-form, and the assumption two simplifies the characteristic exponent dramatically.

The ch.f of the amplitude is (in stationary form)
\Phi_{\delta}(u)=\exp\left(-\frac{\sigma^{\alpha}|u|^{\alpha}}{\omega_1\alpha}\right)
Note that the amplitude process is again alpha-stable.

Sadly, the ch.f of phase flucturation is still very ugly, and can not be expressed explicitly.

張貼者： 於
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