Conditional Monte Carlo for sums, with applications to insurance and finance

Abstract

We consider sums of n i.i.d. random variables with tails close to exp{−x β } for some β > 1. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg & Resnick (1993) are discussed from the point of view of tails rather of densities, using a somewhat different angle, and supplemented with bounds, results on a random number N of terms, and simulation algorithms.

On preemptive-repeat LIFO queues

“…The advantage is, however, that they are much easier implemented than the above exponential tilting scheme. The next two propositions extend results of [2] to more general tails. Proposition 9.3.…”

Monte Carlo estimation of the density of the sum of dependent random variables

“…In the following Sections 3.1 and 3.2 we describe the Conditional Monte Carlo approach [2], as well as an extension of the Asmussen-Kroese estimator. We then use these methods as benchmarks to illustrate the performance of the proposed estimator in various settings.…”

“…Finally, Monte Carlo estimators such as Conditional Monte Carlo [2] and the Asmussen-Kroese estimator [6] utilize details of X's distribution to produce unbiased estimates with a dimension-independent rate of convergence of O(1/n).…”

Computational methods for sums of random variables

“…The following theorem (adapted from Theorem 2.1.2 of [160]) considers how Fourier expansions perform as function approximations: Theorem 1. 13. Given f ∈ L 2 (R, w(x) dx) and the orthonormal functions φ 0 , .…”

“…An advantage of conditional MC is that the variance will either decrease or stay the same relative to the CMC estimator. This technique can achieve a remarkable variance reduction, as exemplified by the Asmussen-Kroese estimator [21]; also, see [13] for a recent review of conditional MC for sums of random variables.…”