As seen in tail risk estimation.

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

where

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:

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, \(\mathbb{P}(L\geq\gamma)\).

TBC.

## Dynamic splitting

TBC.

## Refs

- AsGl07
- Asmussen, S., & Glynn, P. W.(2007) Stochastic Simulation: Algorithms and Analysis. (2007 edition.). New York: Springer
- BoSM17
- botev, zdravko, salome, robert, & mackinlay, D. (2017) Accurate Computation of the Distribution of Sums of Dependent Log-Normals with Applications to the Black-Scholes Model.
- Bote16
- Botev, Z. I.(2016) The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, n/a-n/a. DOI. - BoKr08
- Botev, Z. I., & Kroese, D. P.(2008) An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting.
*Methodology and Computing in Applied Probability*, 10(4), 471–505. DOI. - BoKr12
- Botev, Z. I., & Kroese, D. P.(2012) Efficient Monte Carlo simulation via the generalized splitting method.
*Statistics and Computing*, 22(1), 1–16. DOI. - LaBo15
- Lam, K., & Botev, Z. (2015, November 1) The Dynamic Splitting Method with an application to portfolio credit risk.
- RuKr16
- Rubinstein, R. Y., & Kroese, D. P.(2016) Simulation and the Monte Carlo Method. (3 edition.). Hoboken, New Jersey: Wiley