# Rare-event-conditional estimation

As seen in tail risk estimation.

At the moment I mostly care about splitting simulation, but the set-up for that problem is here.

I consider the problem of simulating some quantity of interest conditional on a tail-event defined on a $$d$$-dimensional continuous random variable $$X$$ and importance function $$S: \mathbb{R}^d\rightarrow \mathbb{R}$$. We write the conditional density $$f^*$$ in terms of the density of $$X$$ as

$f^*(L) = \frac{1}{\ell_\gamma}\mathbb{I}\{L\geq \gamma\}$

where

$\ell(\gamma):=\mathbb{I}\{L\geq \gamma\}$

is a normalising constant; specifically, $$\ell(\gamma)$$ is the cumulative distribution of the random variable $$L.$$

So, using naïve Monte Carlo, you can estimate this by taking the empirical cdf of $$N$$ independent simulations of variable $$L_i$$ as an estimate of the true cdf:

$\hat{\ell}(\gamma) = \frac{1}{N}\sum_{i=1}^N \mathbb{I}\{L_i\geq \gamma\}$

Note that if the quantity of interest is precisely this cdf for values of $$\gamma$$ close to the expectation we may already be done, depending what we regard as a “good” estimate.

But if we care about rare tail events specifically, we probably need to work harder. Suppose hold $$\gamma$$ fixed and $$\ell (\gamma) \ll 10^{-2}$$, we have a bad convergence rate for this estimator. TBC: convergence rates, number of samples.

## Importance sampling

TODO: explain explicitly the variance of this estimator.

I simulate using a different variable $$L'=S(L)$$. I am interested in the probability of a random portfolio loss $$L$$ exceeding a threshold, $$\mathbb{P}(L\geq\gamma)$$.

$\ell(\gamma) := \mathbb{P}\left(S\right)$

TBC.

TBC.