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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.

Methods for Bayesian simulation in the presence of computationally intractable likelihood functions are of growing interest. Termed likelihood-free samplers, standard simulation algorithms such as Markov chain Monte Carlo have been adapted for this setting. In this article, by presenting generalisations of existing algorithms, we demonstrate that likelihood-free samplers can be ambiguous over the form of the target distribution. We also consider the theoretical justification of these samplers. Distinguishing between the forms of the target distribution may have implications for the future development of likelihood-free samplers.

$\alpha$-stable distributions are utilised as models for heavy-tailed noise in many areas of statistics, finance and signal processing engineering. However, in general, neither univariate nor multivariate $\alpha$-stable models admit closed form densities which can be evaluated pointwise. This complicates the inferential procedure. As a result, $\alpha$-stable models are practically limited to the univariate setting under the Bayesian paradigm, and to bivariate models under the classical framework. In this article we develop a novel Bayesian approach to modelling univariate and multivariate $\alpha$-stable distributions based on recent advances in "likelihood-free" inference. We present an evaluation of the performance of this procedure in 1, 2 and 3 dimensions, and provide an analysis of real daily currency exchange rate data. The proposed approach provides a feasible inferential methodology at a moderate computational cost.

We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers. We provide a study of theoretical properties of the new algorithm, and make connections with some existing algorithms. Finally, the sampler is adapted for application under the challenging "likelihood free," approximate Bayesian computation modelling framework, where we demonstrate superior performance over existing likelihood-free samplers.