An Introduction to Stochastic Particle Integration Methods : With Applications to Risk and Insurance

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 compliment alternative approaches adopted in risk management, including closed from 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 in a principled fashion the general framework of interacting particle methods, which goes well beyond the standard particle filtering framework and Sequential Monte Carlo frameworks to instead 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 which we reinterpret under a Feynman-Kac particle integration framework. These functional models are natural mathematical extensions of the traditional change of probability measures, common in designing importance samplers. 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