Zakai Equation Research Articles

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.}

Approximation of the Zakai equation for Diffusions with Noise Correlation and Its Application to Stochastic Volatility Estimation

Approximation of the Zakai equation for Diffusions with Noise Correlation and Its Application to Stochastic Volatility Estimation

Hamilton–Jacobi–Bellman Equations for the Optimal Control of the Duncan–Mortensen–Zakai Equation

We study a class of Hamilton–Jacobi–Bellman (HJB) equations associated to stochastic optimal control of the Duncan–Mortensen–Zakai equation. The equations are investigated in weighted L2 spaces. We introduce an appropriate notion of weak (viscosity) solution of such equations and prove that the value function is the unique solution of the HJB equation. We apply the results to stochastic optimal control problems with partial observation and correlated noise.

Risk sensitive nonlinear filtering with correlated noises

The purpose of this note is to compute risk-sensitive Zakai and Kushne-Stratonovitch equations for partially observed systems with correlated noises. These equations are explicitly solved in the linear exponential quadratic Gaussian case.

Approximation of the Kushner Equation for Nonlinear Filtering

In this paper we discuss the well-posedness and approximation of solutions to the Kushner equation in nonlinear filtering problems. We develop and analyze a time integration method based on the operator splitting. Also, we discuss its relation to the operator-splitting method for the Zakai equation.

A Bayes Formula for Gaussian Noise Processes and its Applications

An elementary approach is used to derive a Bayes-type formula, extending the Kallianpur--Striebel formula for the nonlinear filters associated with the Gaussian noise processes. In the particular cases of certain Gaussian processes, recent results of Kunita and of Le Breton on fractional Brownian motion are derived. We also use the classical approximation of the Brownian motion by the Ornstein--Uhlenbeck dispersion process to solve the instrumentability problem of Balakrishnan. We give precise conditions for the convergence of the filter based on the Ornstein--Uhlenbeck dispersion process to the filter based on the Brownian motion. It is also shown that the solution of the Zakai equation can be approximated by that of a (deterministic) partial differential equation.

Characterization of the optimal filter: the non markov case

We characterize the optimal filter in the nonlinear filtering theory as the unique solution to the Zakai equation. The results are very general as we allow the function h appearing in the filtering model to be discontinuous and unbounded. We consider the standard Markov model as well as the case when the signal X and the observation Y are solutions of a stochastic differential equation where the coefficients are allowed to depend on the past of Y. This is done via some results on existence of stationary solutions and solutions corresponding to certain measure valued evolution equations for controlled (and uncontrolled) martingale problems

Optimal Control and Filtering of the Reproduction Law of a Branching Process

A finite horizon control problem for the reproduction law of a branching process is studied. Some examples with complete information are tackled via the Hamilton–Jacobi–Bellman equation. A partially observable control of the cardinality of the population using the information given by the splitting process is formulated. Though there is correlation between the state and the observations and the observation process has unbounded intensity, a Girsanov-type change of probability measure can be set and the filtering equation for the unnormalized conditional distribution (the Zakai equation) can be derived. Strong uniqueness for the Zakai equation and, as a consequence, also for the Kushner–Stratonovich equation is obtained. A separated control problem is introduced, in which the dynamics are represented by the splitting process and the unnormalized conditional distribution. By the strong uniqueness for the Zakai equation, equivalence between the partially observable control problem and the separated one is proved.

Discrete time nonlinear filtering with marked point process observations

This work discusses the problem of estimating a nonlinear signal process with nonadditive non–Gaussian noise via a marked point process observation. We begin by introducing a modeling of discrete–time marked point processes. In order to derive the exact filter, namely the conditional density of the signal given the observation history, a reference probability is introduced; this probability measure renders the signal and the observation processes independent, as needed. We present a discretetime version of Zakai equation corresponding to our model. By specializing the signal to be linear, we moreover show, under suitable assumptions, that the filter is finite dimensional

Finite Dimensional Filters with Nonlinear Drift, XI: Explicit Solution of the Generalized Kolmogorov Equation in Brockett–Mitter Program

Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late 1970s the concept of the estimation algebra of a filtering system was introduced. Brockett, Clark, and Mitter proposed to use the Wei–Norman approach to solve the nonlinear filtering problem. In 1990, Tam, Wong, and Yau presented a rigorous proof of the Brocket–Mitter program which allows one to construct finite-dimensional recursive filters from finite–dimensional estimation algebras. Later Yau wrote down explicitly a system of ordinary differential equations and generalized Kolmogorov equation to which the robust Duncan–Mortenser– Zakai equation can be reduced. Thus there remains three fundamental problems in Brockett–Mitter program. The first is the problem of finding explicit solution to the generalized Kolmogorov equation. The second is the problem of finding real-time solution of a system of ODEs. The third is the Brockett's problem of classification of finite–dimensional estimation algebras. In this paper, we solve the first problem.

Convergence of a Branching Particle Method to the Solution of the Zakai Equation

We construct a sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation. The algorithm based on this result can be used to solve numerically the filtering problem. The result is an improvement of the one presented in a recent paper [Crisan and T. Lyons, Prob. Theory Related Fields, 109 (1997), pp. 217--244], because it eliminates the extra degree of randomness introduced there.

Approximate Filter for the Conditional Law of a Partially Observed Process in Nonlinear Filtering

We study a problem of singular perturbations for a special class of nonlinear filtering problems in which the dimension of the signal process is 2 and only one of the two components of this process is observed. We propose an approximate filter of finite dimension for the observed part. Using this filter, we construct an approximate filter of infinite dimension for the nonobserved part. This filter solves a Zakai-type equation whose spatial variable dimension is 1 even though the spatial variable dimension of the Zakai equation solved by the exact filter is 2. The method used gives the order of the approximation error.

Classes of Nonlinear Partially Observable Stochastic Optimal Control Problems with Explicit Optimal Control Laws

This paper introduces certain nonlinear partially observable stochastic optimal control problems which are equivalent to completely observable control problems with finite-dimensional state space. In some cases the optimal control laws are analogousto linear-exponential-quadratic-Gaussian and linear-quadratic-Gaussian tracking problems. The problems discussed allow nonlinearities to enter the unobservable dynamics as gradients of potential functions. The methodology is based on explicit solutions of a modified Duncan--Mortensen--Zakai equation.

Reduction of the Zakai equation by invariance group techniques

A general procedure, inspired from that used for deterministic partial differential equations, is presented to reduce the Zakai stochastic Pde of filtering on R n to a stochastic Pde on a lower-dimensional space R m , with m < n. The method is based upon invariance group techniques. We show how the existence of invariant solutions of the Zakai equation is related to geometric properties of the infinitesimal generator of the signal process. An illustration of the method to a two-dimensional tracking problem with bearings-only measurements is presented. With a specific choice of the bearings-dependent output function, we obtain a continuous model for which the Zakai equation has solutions which can be computed from a one-dimensional stochastic Pde instead of a two-dimensional Pde for the general solution.

Zakai equations for hilbert space valued processes*

A state process is described by either a discrete time Hilbert space valued process, or a stochastic differential equation in Hilbert space. The state is observed through a finite dimensional process. Using a change of measure and a Fusive theorem the Zakai equation is obtained in discrete or continuous time. A risk sensitive state estimate is also defined

Existence and decay estimates for time dependent parabolic equation with application to Duncan–Mortensen–Zakai equation

Existence and decay estimates for time dependent parabolic equation with application to Duncan–Mortensen–Zakai equation

Nonlinear filtering and measure-valued processes

We construct a sequence of branching particle systems with time and space dependent branching mechanisms whose expectation converges to the solution of the Zakai equation. This gives an alternative numerical method to solve the Filtering Problem.

Nonlinear filtering of a system of logistic equations

This paper is concerned with the filtering problem for a nonlinear stochastic system of prey-predator logistic equations. Based on the innovations approach, we establish the Zakai equation for the unnormalised conditional distribution and the adjoint Zakai equation for the unnormalised conditional density of the nonlinear filter. Using a perturbation technique, we obtain the appropriate expressions for the unnormalised conditional distribution and density of stochastic integrals with respect to the observation processes.

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained.

A powerful numerical technique solving Zakai equation for nonlinear filtering

In this paper we have developed a simple but powerful numerical method forthe approximation of the unnormalized conditional (probability) density offiltered diffusion process which satisfies Zakai equation and solves thenonlinear filtering problem. Using Galerkin technique the solution of Zakaiequation is approximated by means of a sequence of nonstandard basisfunctions given by a parameterized family of Gaussian densities. The methodis then illustrated by some examples.