Filtering the Wright-Fisher diffusion

Cogent Mathematics, 2016

The problem of filtering of unobservable components x(t) of a multidimensional continuous diffusion Markov process z(t) = (x(t), y(t)), given the observations of the (multidimensional) process y(t) taken at discrete consecutive times with small time steps, is analytically investigated. On the base of that investigation the new algorithms for simulation of unobservable components, x(t), and the new algorithms of nonlinear filtering with the use of sequential Monte Carlo methods, or particle filters, are developed and suggested. The analytical investigation of observed quadratic variations is also developed. The new closed-form analytical formulae are obtained, which characterize dispersions of deviations of the observed quadratic variations and the accuracy of some estimates for x(t). As an illustrative example, estimation of volatility (for the problems of financial mathematics) is considered. The obtained new algorithms extend the range of applications of sequential Monte Carlo methods, or particle filters, beyond the hidden Markov models and improve their performance.

arXiv: Mathematical Physics, 2017

Filtering theory gives an explicit models for the flow of information and thereby quantifies the rates of change of information supplied to and dissipated from the filter's memory. Here we extend the analysis of Mitter and Newton from linear Gaussian models to general nonlinear filters involving Markov diffusions.The rates of entropy production are now generally the average squared-field (co-metric) of various logarithmic probability densities, which may be interpreted as Fisher information associate with Gaussian perturbations (via de Bruijn's identity). We show that the central connection is made through the Mayer-Wolf and Zakai Theorem for the rate of change of the mutual information between the filtered state and the observation history. In particular, we extend this Theorem to cover a Markov diffusion controlled by observations process, which may be interpreted as the filter acting as a Maxwell's Daemon applying feedback to the system.

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2008

In this paper we introduce a novel particle filter scheme for a class of partiallyobserved multivariate diffusions. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike currently available methods, our particle filters do not require approximations of the transition and/or the observation density using time-discretisations. Instead, they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in . We introduce the Generalised Poisson Estimator, which generalises the Poisson Estimator of . A central limit theorem is given for our particle filter scheme.

1988

The paper presents some recently obtained results on the nonlinear filtering problem for infinite dimensional processes. The optimal filter is obtained as the unique solution of certain measure valued equations. Robustness properties — both pathwise and statistical — are given and a preliminary result shows consistency with the stochastic calculus theory. Applications to random fields and models of voltage potential in neurophysiology are briefly discussed.

EURASIP Journal on Advances in Signal Processing, 2018

This paper introduces a new algorithm to approximate smoothed additive functionals of partially observed diffusion processes. This method relies on a new sequential Monte Carlo method which allows to compute such approximations online, i.e., as the observations are received, and with a computational complexity growing linearly with the number of Monte Carlo samples. The original algorithm cannot be used in the case of partially observed stochastic differential equations since the transition density of the latent data is usually unknown. We prove that it may be extended to partially observed continuous processes by replacing this unknown quantity by an unbiased estimator obtained for instance using general Poisson estimators. This estimator is proved to be consistent and its performance are illustrated using data from two models.

Stochastic Processes and their Applications, 2006

In this paper, we study the problem of estimating a Markov chain X(signal) from its noisy partial information Y , when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{X n |Y n , . . . , Y 1 }, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a nondynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.

Statistical Methods for Stochastic Differential Equations …, 2010

In this chapter we consider parametric inference based on discrete time observations X0, Xt1,..., Xtn from a d-dimensional stochastic process. In most of the chapter the statistical model for the data will be a diffusion model given by a stochastic differential equation. We ...

Probability Theory and Related Fields, 2001

2009

For a wide class of continuous-time Markov processes, including all irreducible hypoelliptic diffusions evolving on an open, connected subset of R d , the following are shown to be equivalent:

Latin American Journal of Probability and Mathematical Statistics, 2011

We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the data and construct a nonparametric estimator of the invariant density of the process. In the second step, which is referred to as a matching step, we exploit a characterisation of the invariant density as a solution of a certain ordinary differential equation, replace the invariant density in this equation by its nonparametric estimator from the smoothing step in order to arrive at an intuitively appealing criterion function, and next define our estimator of the parameter of interest as a minimiser of this criterion function. In many interesting examples such an estimator will be computationally less intense than the more conventional estimators obtained through approximation of the likelihood function associated with the observations. Our main result shows that our estimator is √ n-consistent under suitable conditions. We also discuss a way of improving its asymptotic performance through a one-step Newton-Raphson type procedure.