I am given a diffusion with a local volatility to price barrier options:
$$dX(t)=X(t)\mu dt+X(t)\sigma(t,X)dW_t$$
I want to use Importance Sampling to price barrier options "far" out of the money. I did some research and found https://pdfs.semanticscholar.org/4fe5/94e3c7667c762cf1f7d841fcd0a4bf30f255.pdf
This is the simplest I found about the subject as I am looking to simply implement the method.
However, I am struggling to use this in practice. From my understand one can use (with the same notation as in the paper)
$$V \approx V_g = E_g[G(X)\frac{f(X)}{g(X)}]$$
instead of using:
$$V \approx V_f = E_f[G(X)]$$
with $f$ the original density and $g$ the newly define one that minimises the variance of the monte carlo estimator.
If $f$ is given by : $$f(x)=(2 \pi)^{-\frac{n}{2}} \mathrm{e}^{-\frac{1}{2} x^{T} x}$$
then $g$ can be defined by: $$ g_{\mu}(x)=(2 \pi)^{-\frac{n}{2}} \mathrm{e}^{-\frac{1}{2}(x-\mu)^{T}(x-\mu)}$$
with $\mu$: $$\min _{\mu} E_{f}\left[G^{2}(X) \frac{f(X)}{g_{\mu}(X)}\right]$$
How can I practicaly use this in my Monte Carlo simulation, for the $X(t)$-process given above?