I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf
This is a paper by Nengjiu Ju who uses talyor series to express higher moments of multidimensional gaussian distributions in an one-dimensional expression.
Equation (9) in the paper is baffling me at the moment. Hope you could shed the light of it. Here is the equation.
\begin{eqnarray} \mathbb{E}[e^{i\phi X(z)}]=\mathbb{E}[e^{i\phi Y(z)}]{\mathbb{E}(e^{i\phi X(z)}) \over \mathbb{E}(e^{i\phi Y(z)})}=\mathbb{E}[e^{i\phi Y(z)}]f(z) \end{eqnarray} Here $Y(z)$ is a normal random variable with mean $m(z^2)$ and variance $v(z^2 ).$ $X(z) = \log(A(z)), $ $A(z)=\sum_{i=1}^N x_iS_ie^{(g_i−0.5z^2\sigma_i^2)T+z\sigma_iw_i(T)},$ where $g_i, \sigma_i, w_i, x_i$ are constants.
How does the author say ${\mathbb{E}(e^{i\phi X(z)}) \over \mathbb{E}(e^{i\phi Y(z)})}=f(z),$ where $f(z)$ is the density function of $X(z)$?