Date: Wednesday 06 April 2017
Time: 12:45 - 13:30 (Lunch available from 12:35)
Room: RA 1501 (Ravelijn)
Title: Error bounds and approximations for heavy-tailed stochastic systems
Abstract:
Many random phenomena are driven by data characterised by asymmetry and thick tails. A typical example is insurance loss data, which are known to be heavy-tailed and oftentimes non-negative, right-skewed and leptokurtic. In this case, accurately fitting the tail is an important modelling task, as the losses in the tail, though rare in frequency, are the ones that have the most impact on the operations of an insurer, possibly leading to bankruptcy. Similar concerns dominate other stochastic systems with heavy-tailed input.
At the same time, exact analysis of systems with heavy-tailed input is typically challenging. The usual route is to develop suitable approximations. One way to do so is to approximate the heavy-tailed data with a distribution more suitable for computations. The typical class of distributions chosen is the phase-type distributions, as this class has two main advantages. First, it can be used to approximate any distribution on a positive support arbitrarily closely and second, it introduces a structure in the model that allows for the usage of powerful computational (iterative) techniques known as matrix-analytic methods. Known disadvantages of various approximation schemes for heavy-tailed data are the quality of the approximation for small or large values of the support or technical requirements, such as finite higher moments, which may impose unnecessary restrictions.
In this talk, we present error bounds and accurate approximations for stochastic systems with heavy-tailed input. Using ideas from spectral theory and perturbation analysis, we combine desirable characteristics of the main approximation directions while avoiding their most glaring disadvantages. The approximations we discuss maintain the computational tractability of phase-type approximations, capture the correct tail behaviour, do not require finite higher-order moments, and provide small absolute and relative errors, independent of the initial conditions.