A Direct Approach to Deriving Filtering Equations for Diffusion Processes

Filtering equations are derived for conditional probability density functions in case of partially observable diffusion processes by using results and methods from the L _p -theory of SPDEs. The method of derivation is new and does not require any knowledge of filtering theory. Accepted 31 July 2000. Online publication 13 November 2000.     

Filtering of a Markov Jump Process with Counting Observations

This paper concerns the filtering of an R ^d -valued Markov pure jump process when only the total number of jumps are observed. Strong and weak uniqueness for the solutions of the filtering equations are discussed. Accepted 12 November 1999     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

Robust Approximation in a Filtering Problem with Real State Space and Counting Observations

Let (X _t ,Y _t ) be a pure jump Markov process, where X _t takes values in \bf R and Y _t is a counting process. We compare the filter of this system and a filter of a suitably modified system. We compute an explicit bound for the distance in the so-called bounded Lipschitz metric between the two filters. Finally we show how to use this bound to construct a discrete space approximation of the filter. Accepted 7 December 1999     

Counting Observations: A Note on State Estimation Sensitivity with an L 1 -Bound

Let (X _t , Y _t ) be a pure jump Markov process: the state X _t takes real values and the observation Y _t is a counting process. The two processes are allowed to have common jump times. Let ϕ(X(⋅)) be a functional of the state trajectory restricted to the time interval [0, T] . If we change the infinitesimal parameters and/ or the initial distribution, then we introduce an error in computing the conditional law of ϕ(X(⋅)) given the observation up to time T . In this paper we give an explicit L ¹ -bound for this error. Accepted 9 March 2001. Online publication 20 June 2001.     

Envelopes of Sets of Measures, Tightness, and Markov Control Processes

We introduce upper and lower envelopes for sets of measures on an arbitrary topological space, which are then used to give a tightness criterion. These concepts are applied to show the existence of optimal policies for a class of Markov control processes. Accepted 22 May 1998     

On Risk-Sensitive Ergodic Impulsive Control of Markov Processes

Impulsive control of continuous-time Markov processes with risk- sensitive long-run average cost is considered. The most general impulsive control problem is studied under the restriction that impulses are in dyadic moments only. In a particular case of additive cost for impulses, the impulsive control problem is solved without restrictions on the moments of impulses. Accepted 30 April 2001. Online publication 29 August 2001.     

The rank of the covariance matrix of an evanescent field

Evanescent random fields arise as a component of the 2D Wold decomposition of homogeneous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space-time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem.     

Filtering with Discrete State Observations

The problem of estimating a finite state Markov chain observed via a process on the same state space is discussed. Optimal solutions are given for both the ``weak' and ``strong' formulations of the problem. The ``weak' formulation proceeds using a reference probability and a measure change for the Markov chain. The ``strong' formulation considers an observation process related to perturbations of the counting processes associated with the Markov chain. In this case the ``small noise' convergence is investigated. Accepted 7 April 1998     

Adaptive Control of a Partially Observed Discrete Time Markov Process

An adaptive control problem of a discrete time Markov process that is completely observed in a fixed recurrent domain and is partially observed elsewhere is formulated and a solution is given by constructing an approximately self-optimal strategy. The state space of the Markov process is either a closed subset of Euclidean space or a countable set. Another adaptive control problem is solved where the process is always only partially observed but there is a family of random times when the process evaluated at these times is a family of independent, identically distributed random variables. Accepted 26 April 1996     

Markov Decision Processes with Distribution Function Criterion of First-Passage Time

In this paper we discuss MDP with distribution function criterion of first-passage time. Some properties of several kinds of optimal policies are given. Existence results and algorithms for these optimal policies are given in this paper. Accepted 24 July 2000. Online publication 12 April 2001.     

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.     

Arbitrage Opportunities for a Class of Gladyshev Processes

Geometric versions of a class of Gaussian processes are investigated as possible models for stock prices. Arbitrage opportunities are constructed for these processes showing that option pricing is not possible with these models. Accepted 30 March 1999     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

Fluctuations of interacting Markov chain Monte Carlo methods

We present a multivariate central limit theorem for a general class of interacting Markov chain Monte Carlo algorithms used to solve nonlinear measure-valued equations. These algorithms generate stochastic processes which belong to the class of nonlinear Markov chains interacting with their empirical occupation measures. We develop an original theoretical analysis based on resolvent operators and semigroup techniques to analyze the fluctuations of their occupation measures around their limiting values.     

On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen’s inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E subset(-1,1) be a compact set, let μ be a positive Borel measure with support supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function frac{1}{2π i} ∈t_E frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Qnot equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b frac{log(d μ/ d x)}{sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

Robust estimation in a nonlinear cointegration model

This paper considers the nonparametric M-estimator in a nonlinear cointegration type model. The local time density argument, which was developed by Phillips and Park (1998) [6] and Wang and Phillips (2009) [9], is applied to establish the asymptotic theory for the nonparametric M-estimator. The weak consistency and the asymptotic distribution of the proposed estimator are established under mild conditions. Meanwhile, the asymptotic distribution of the local least squares estimator and the local least absolute distance estimator can be obtained as applications of our main results. Furthermore, an iterated procedure for obtaining the nonparametric M-estimator and a cross-validation bandwidth selection method are discussed, and some numerical examples are provided to show that the proposed methods perform well in the finite sample case.