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 -dimensional continuous random variable and importance function . We write the conditional density in terms of the density of as
is a normalising constant; specifically, is the cumulative distribution of the random variable
So, using naïve Monte Carlo, you can estimate this by taking the empirical cdf of independent simulations of variable as an estimate of the true cdf:
Note that if the quantity of interest is precisely this cdf for values of 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 fixed and , we have a bad convergence rate for this estimator. TBC: convergence rates, number of samples.
TODO: explain explicitly the variance of this estimator.
I simulate using a different variable . I am interested in the probability of a random portfolio loss exceeding a threshold, .
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