Can you explain me how can I compute the gauss hermite quadrature using a log normal distribution? Suppose $r \sim N (\mu, \sigma)$ and then $y=e^r \sim LN(\mu, \sigma)$, how can I compute $\mathbb{E}[h(y)]$ given a general function h(.) and the Gauss-Hermite Quadrature? Basically, I would like to redo the same example of https://en.wikipedia.org/wiki/Gauss%E2%80%93Hermite_quadrature with a log normal distribution.
Thanks a lot for any help.
In the linked Wikipedia example, they show how to numerically compute the expectation of $h(Y)$ for some function $h$ if $Y\sim N(\mu,\sigma^2)$:
\begin{equation} E[h(Y)] = \int_{-\infty}^{+\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(y-\mu)^2}{2\sigma^2} \right) h(y) dy \approx \frac{1}{\sqrt{\pi}} \sum_{i=1}^n w_i h(\sqrt{2} \sigma x_i + \mu) \end{equation}
where the $w_i$ and $x_i$'s are standard Gauss-Hermite weights and nodes. (Note the parameterization is in terms of the variance $\sigma^2$ for consistency with the reference.)
Suppose you instead have a function $g$ on $[0,\infty)$, and want to know $E[g(Z)]$ for lognormal random variable $Z\sim LN(\mu,\sigma^2)$, meaning that $\log(Z) \sim N(\mu,\sigma^2)$. The solution is to define the function $h(y) = g(\exp(y))$, and use Gauss-Hermite as above on this new $h$.