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 naĂŻve 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 â€śgoodâ€ť 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. đźš§ convergence rates, number of samples.

## Importance sampling

đźš§ 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.

## Dynamic splitting

TBC.

# Refs

Asmussen, SĂ¸ren, and Peter W. Glynn. 2007. *Stochastic Simulation: Algorithms and Analysis*. 2007 edition. New York: Springer.

Ben Rached, Nadhir, Zdravko Botev, Abla Kammoun, Mohamed-Slim Alouini, and Raul Tempone. 2018. â€śOn the Sum of Order Statistics and Applications to Wireless Communication Systems Performances.â€ť *IEEE Transactions on Wireless Communications* 17 (11): 7801â€“13. https://doi.org/10.1109/TWC.2018.2871201.

Botev, Zdravko I., and Dirk P. Kroese. 2008. â€śAn Efficient Algorithm for Rare-Event Probability Estimation, Combinatorial Optimization, and Counting.â€ť *Methodology and Computing in Applied Probability* 10 (4): 471â€“505. https://doi.org/10.1007/s11009-008-9073-7.

â€”â€”â€”. 2012. â€śEfficient Monte Carlo Simulation via the Generalized Splitting Method.â€ť *Statistics and Computing* 22 (1): 1â€“16. https://doi.org/10.1007/s11222-010-9201-4.

Botev, Zdravko I., Robert Salomone, and Daniel Mackinlay. 2019. â€śFast and Accurate Computation of the Distribution of Sums of Dependent Log-Normals.â€ť *Annals of Operations Research*, February. https://doi.org/10.1007/s10479-019-03161-x.

Botev, Zdravko, and Pierre Lâ€™Ecuyer. 2017. â€śSimulation from the Normal Distribution Truncated to an Interval in the Tail.â€ť In *Proceedings of the 10th EAI International Conference on Performance Evaluation Methodologies and Tools*, 23â€“29. VALUETOOLSâ€™16. ICST, Brussels, Belgium, Belgium: ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering). https://doi.org/10.4108/eai.25-10-2016.2266879.

Botev, Z. I. 2017. â€śThe Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting.â€ť *Journal of the Royal Statistical Society: Series B (Statistical Methodology)* 79 (1): 125â€“48. https://doi.org/10.1111/rssb.12162.

CĂ©rou, F., P. Del Moral, T. Furon, and A. Guyader. 2011. â€śSequential Monte Carlo for Rare Event Estimation.â€ť *Statistics and Computing* 22 (3): 795â€“808. https://doi.org/10.1007/s11222-011-9231-6.

Johansen, Adam M., Pierre Del Moral, and Arnaud Doucet. 2006. â€śSequential Monte Carlo Samplers for Rare Events.â€ť In *Proceedings of the 6th International Workshop on Rare Event Simulation*, 256â€“67. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.2888.

Rubinstein, Reuven Y., and Dirk P. Kroese. 2016. *Simulation and the Monte Carlo Method*. 3 edition. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.