Inference for a class of partially observed point process models

Abstract

This paper presents a simulation-based framework for sequential inference from partially and discretely observed point process (PP's) models with static parameters. Taking on a Bayesian perspective for the static parameters, we build upon sequential Monte Carlo (SMC) methods, investigating the problems of performing sequential filtering and smoothing in complex examples, where current methods often fail. We consider various approaches for approximating posterior distributions using SMC.Our approaches, with some theoretical discussion are illustrated on a doubly stochastic point process applied in the context of finance. Some

Multilevel sequential Monte Carlo samplers

“…This follows from standard calculations in the analysis of Feynman-Kac formulae; see e.g. the proof of Proposition 2 in [16]. For (ii).This follows from [6, Lemma 7.3.3] and (i).…”

Static-parameter estimation in piecewise deterministic processes using particle Gibbs samplers

“…In anticipation of Subsection 5.2 we note here that the variance-reduction techniques mentioned therein cannot be applied to such degenerate problems which makes particleGibbs-sampler-based inference impractical. However, -based algorithms, such as those from Centanni and Minozzo (2006a,b); Del Moral et al (2007); Martin et al (2012) will not be practical for such models either and the problem remains generally unsolved.…”

“…A stochastic expectation-maximisation algorithm based on a reversible-jump ( ) sampler (Green, 1995) was introduced by Centanni and Minozzo (2006a,b). A simple sampler was attempted in Del Moral et al (2007) to which some improvements were made in Martin et al (2012). In addition, Rao and Teh (2012) developed a Gibbs sampler for the special case in which the state space is discrete.…”

“…(1) We provide new insight into the approximation induced by the lter for and by related algorithms used in Del Moral et al (2006Moral et al ( , 2007; Martin et al (2012). We also suggest a way of ensuring the existence of the importance weights.…”

Recursive Backward Scheme for the Solution of a Bsde With a Non Lipschitz Generator

“…Of course, it is not known in advance when and of which amount the price adjustments will be. Processes of that kind have been used, for example, in the field of financial modeling and filtering, between others, by Engle and Russell [12], Centanni and Minozzo [10], Gerardi and Tardelli [15] and Martin, Jasra, and McCoy [24]. Furthermore, regarding microstructure analysis see Bacry et al [1], and for option pricing see Pringent [25], Frey and Runggaldier [14], and Cartea [8].…”