An introduction to particle integration methods: with applications to risk and insurance

Interacting particle methods are increasingly used to sample from complex and high-dimensional distributions. These stochastic particle integration techniques can be interpreted as an universal acceptance-rejection sequential particle sampler equipped with adaptive and interacting recycling mechanisms. Practically, the particles evolve randomly around the space independently and to each particle is associated a positive potential function. Periodically, particles with high potentials duplicate at the expense of low potential particle which die. This natural genetic type selection scheme appears in numerous applications in applied probability, physics, Bayesian statistics, signal processing, biology, and information engineering. It is the intention of this paper to introduce them to risk modeling. From a purely mathematical point of view, these stochastic samplers can be interpreted as Feynman-Kac particle integration methods. These functional models are natural mathematical extensions of the traditional change of probability measures, commonly used to design an importance sampling strategy. In this article, we provide a brief introduction to the stochastic modeling and the theoretical analysis of these particle algorithms. Then we conclude with an illustration of a subset of such methods to resolve important risk measure and capital estimation in risk and insurance modelling.