On hidden Markov chains and finite stochastic systems

In this paper we study various properties of finite stochastic systems or hidden Markov chains as they are alternatively called. We discuss their construction following different approaches and we also derive recursive filtering formulas for the different systems that we consider. The key tool is a simple lemma on conditional expectations

The Dynamics of Retail Oligopolies

Dynamic Models, Retail Indusry, Markov perfect equilibrium, Oligopoly

Power vs. Spectrum 2-D Sensing in Energy Harvesting Cognitive Radio Networks

Energy harvester based cognitive radio is a promising solution to address the shortage of both spectrum and energy. Since the spectrum access and power consumption patterns are interdependent, and the power value harvested from certain environmental sources are spatially correlated, the new power dimension could provide additional information to enhance the spectrum sensing accuracy. In this paper, the Markovian behavior of the primary users is considered, based on which we adopt a hidden input Markov model to specify the primary vs. secondary dynamics in the system. Accordingly, we propose a 2-D spectrum and power (harvested) sensing scheme to improve the primary user detection performance, which is also capable of estimating the primary transmit power level. Theoretical and simulated results demonstrate the effectiveness of the proposed scheme, in term of the performance gain achieved by considering the new power dimension. To the best of our knowledge, this is the first work to jointly consider the spectrum and power dimensions for the cognitive primary user detection problem

The iterated auxiliary particle filter

We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions psi and has an associated psi-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence psi* that is optimal in the sense that the psi*-auxiliary particle filter's estimate of L has zero variance. In practical applications, psi* is unknown so the psi*-auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate psi*, and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm

MCMC, likelihood estimation and identifiability problems in DLM models

In this article we deal with the identification problem within the Dynamic Linear Models family and show that using Bayesian estimation procedures we can deal better with these problems in comparison with the traditional Maximum Likelihood estimation approach. Using a Bayesian approach supported by Markov chain Monte Carlo techniques, we obtain the same results as the Maximum likelihood approach in the case of identifiable models, but in the case of non-identifiable models, we were able to estimate the parameters that are identifiable, as well as to pinpoint the troublesome parameters. Assuming a Bayesian approach, we also discuss the computational aspects, namely the ongoing discussion between single- versus multi-move samplers. Our aim is to give a clear example of the benefits of adopting a Bayesian approach to the estimation of high dimensional statistical models.Bayesian Statistics, DLM Models, Markov chain Monte Carlo, Maximum Likelihood, Model Identification.

EM-algorithm in software reliability modeling

International audienceThe purpose of this paper is to give an overview on the use of the Expectation-Maximization (EM) algorithm in software reliability modeling. This algorithm is related to Maximum Likelihood Estimates (MLE) of parameters in a context of missing data. Different ways to implement this algorithm are highlighted for hidden Markov models in software reliability

USLV: Unspanned Stochastic Local Volatility Model

We propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Our model extends the linearity-generating unspanned volatility term structure model by Carr et al. (2011) by adding a local volatility layer to it. We outline efficient numerical schemes for pricing derivatives in this framework for a particular four-factor specification (two "curve" factors plus two "volatility" factors). We show that the dynamics of such a system can be approximated by a Markov chain on a two-dimensional space (Z_t,Y_t), where coordinates Z_t and Y_t are given by direct (Kroneker) products of values of pairs of curve and volatility factors, respectively. The resulting Markov chain dynamics on such partly "folded" state space enables fast pricing by the standard backward induction. Using a nonparametric specification of the Markov chain generator, one can accurately match arbitrary sets of vanilla option quotes with different strikes and maturities. Furthermore, we consider an alternative formulation of the model in terms of an implied time change process. The latter is specified nonparametrically, again enabling accurate calibration to arbitrary sets of vanilla option quotes.Comment: Sections 3.2 and 3.3 are re-written, 3 figures adde