An Introduction to Stochastic Particle Integration Methods: With Applications to Risk and Insurance
Résumé
This article presents a guided introduction to a general class of interacting particle methods and explains throughout how such methods may be adapted to solve general classes of inference problems encountered in actuarial science and risk management. Along the way, the resulting specialized Monte Carlo solutions are discussed in the context of how they complemented alternative approaches adopted in risk management, including closed form bounds and asymptotic results for functionals of tails of risk processes.
The development of the article starts from the premise that whilst interacting particle methods are increasingly used to sample from complex and high-dimensional distributions, they have yet to be generally adopted in inferential problems in risk and insurance. Therefore, we introduce a range of methods which can all be interpreted in the general framework of interacting particle methods, which goes well beyond the standard particle filtering framework and Sequential Monte Carlo frameworks. For the applications we consider in risk and insurance we focus on particular classes of interacting particle genetic type algorithms. These stochastic particle integration techniques can be interpreted as a universal acceptance-rejection sequential particle sampler equipped with adaptive and interacting recycling mechanisms. We detail how one may reinterpret these stochastic particle integration techniques under a Feynman-Kac particle integration framework. In the process, we illustrate how such frameworks act as natural mathematical extensions of the traditional change of probability measures, common in designing importance samplers for risk managements applications.