Levy过程

Levy测度

特征函数

MJ

${\lambda }_t\frac{1}{\sqrt{2\text{π}}{\sigma }_J}\exp \left\{-\frac{{\left(x-{\mu }_J\right)}^2}{2{\sigma }_J^2}\right\}$

$\exp \left\{\lambda \left({\text{e}}^{i\alpha u-\frac{1}{2}{\delta }^2{u}^2}-1\right)\right\}$

KOU

$\{\begin{array}{l}p\lambda {\eta }_1{\text{e}}^{-{\eta }_{1}x}x>0\\ \left(1-p\right)\lambda {\eta }_2{\text{e}}^{{\eta }_{2}x}x<0\end{array}$

$\exp \left\{t\left[i\mu u-\frac{1}{2}{\sigma }^2{u}^2+iu\lambda \left(\frac{p}{{\eta }_1-iu}-\frac{1-p}{{\eta }_2+iu}\right)\right]\right\}$

NIG

${\text{e}}^{\beta x}\frac{\delta \alpha }{\text{π}\left|x\right|}{K}_1\left(\alpha \left|x\right|\right)$

$\exp \left\{-\delta \left(\sqrt{{\alpha }^2-{\left(\beta +iu\right)}^2}-\sqrt{{\alpha }^2-{\beta }^2}\right)\right\}$

VG

$\{\begin{array}{l}\frac{C{\text{e}}^{Gx}}{\left|x\right|},x<0\\ \frac{C{\text{e}}^{-Mx}}{x},x>0\end{array}$

${\left(\frac{GM}{GM+\left(M-G\right)\text{i}u+u^2}\right)}^C$

α stable

$\{\begin{array}{l}\frac{C_{+}}{x^{\alpha +1}},x>0\\ \frac{C_{-}}{{\left|x\right|}^{\alpha +1}},x<0\end{array}$

$\{\begin{array}{l}\exp \left\{-{\sigma }^{\alpha }{\left|\theta \right|}^{\alpha }\left(1-i\beta \left(sign\left(\theta \right)\tan \frac{\text{π}\alpha }{2}\right)\right)+i\mu \theta \right\},\alpha \ne 1\\ \exp \left\{-{\sigma }^{\alpha }\left|\theta \right|\left(1+i\beta \frac{2}{\text{π}}\left(sign\left(\theta \right)\ln \left|\theta \right|\right)\right)+i\mu \theta \right\},\alpha =1\end{array}$