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. 🏗 convergence rates, number of samples.

Importance sampling

🏗 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.

Large deviations

The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic system is observed, the amplitude of the noise perturbing a dynamical system or the temperature of a chemical reaction. The theory has applications in many different scientific fields, ranging from queuing theory to statistics and from finance to engineering. It is also increasingly used in statistical physics for studying both equilibrium and nonequilibrium systems. In this context, deep analogies can be made between familiar concepts of statistical physics, such as the entropy and the free energy, and concepts of large deviation theory having more technical names, such as the rate function and the scaled cumulant generating function.

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