Nonlinear Filtering Equations Research Articles

Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

A class of non-parametric statistical manifolds modelled on Sobolev space

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on {mathbb {R}}^d, each containing a smoothly embedded submanifold of probability measures. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert–Sobolev spaces and mixed-norm spaces, and support the Fisher–Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this “lifts” to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W^{2,1}; i.e. it maps each of these spaces continuously into itself. In contrast with non-parametric exponential manifolds, the density itself belongs to the model space, and the range of the chart is the whole of this space. Some of the results make essential use of a log-Sobolev embedding theorem, which also sharpens existing results concerning the regularity of statistical divergences on the manifolds. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker–Planck equation) are outlined.

Generation of Conditional Densities in Nonlinear Filtering for Infinite-Dimensional Systems

Consider a signal process that takes values in an infinite-dimensional space and assume that this process is observed through a nonlinear diffusion process. In the nonlinear filtering theory literature, recursive expressions for such problems in the form of conditional expectations and, consequently, measure-valued stochastic partial differential equations on the underlying infinite-dimensional space are obtained. In this work, state estimation problems for such stochastic systems are considered, and nonlinear filtering equations in the form of conditional densities are obtained. This is achieved by first deriving the filtering equations in the form of conditional expectations for a Banach space-valued state process whose dynamics depends on a finite-dimensional Brownian motion. Next, a recursion for the conditional density, which we assume to exist with respect to a $\sigma$ -finite measure, is obtained. For such an analysis, we obtain the adjoint operator for the generator of the state process, which requires to use the theory of integration by parts in the infinite-dimensional domains. Finally, we discuss some potential application of the derived filtering equations in a portfolio optimization problem for portfolios, which consist of bonds whose price curve is interpreted as a stochastic partial differential equation taking values in a Banach space. Applications in mean field games are also demonstrated.

Uniqueness for measure-valued equations of nonlinear filtering for stochastic dynamical systems with Lévy noise

Abstract In the paper we study the Zakai and Kushner–Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner–Stratonovich equation is unique in joint law.

Mean Field Game Theory with a Partially Observed Major Agent

Mean field game (MFG) theory where there is a major agent and many minor agents (MM-MFG) has been formulated for both the linear quadratic Gaussian (LQG) case and for the case of nonlinear state dynamics and nonlinear cost functions. In this framework, even asymptotically (as the population size $N$ approaches infinity), and in contrast to the situation without major agents, the mean field term becomes stochastic due to the stochastic evolution of the state of the major agent; furthermore, the best response control actions of the minor agents depend on the state of the major agent as well as the stochastic mean field. In a decentralized environment, one is led to consider the situation where the agents are provided only with partial information on the major agent's state; in this work such a scenario is considered for systems with nonlinear dynamics and cost functions, and an $\epsilon$-Nash MFG theory is developed for this MM-MFG setup. The approach to the problem of partially observed MM-MFG systems adopted in this work is to follow the procedure of constructing the associated completely observed system via the application of nonlinear filtering theory; consequently, as a first step, nonlinear filtering equations are obtained for partially observed stochastic dynamical systems whose state equations contain a measure term corresponding to the distribution of the solution of a state process. Stochastic control theory for systems with random parameters is next generalized to the partially observed case by lifting the analysis to the infinite-dimensional domain. To achieve this, the Itô--Kunita lemma is first generalized to processes taking values in a subset of $L^1$ consisting of the space of solutions of the conditional density process generated by the filtering equations. The existence and uniqueness of solutions to the MFG system of equations is next established by a fixed point argument in the Wasserstein space of random probability measures in which the robustness property of nonlinear filtering theory is used. Finally, the $\epsilon$-Nash property of such a solution is analyzed in this setting where the state consists of finite- and infinite-dimensional (density) valued stochastic processes.

A Kushner–Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification

A Monte Carlo filter, based on the idea of averaging over characteristics and fashioned after a particle-based time-discretized approximation to the Kushner–Stratonovich (KS) nonlinear filtering equation, is proposed. A key aspect of the new filter is the gain-like additive update, designed to approximate the innovation integral in the KS equation and implemented through an annealing-type iterative procedure, which is aimed at rendering the innovation (observation–prediction mismatch) for a given time-step to a zero-mean Brownian increment corresponding to the measurement noise. This may be contrasted with the weight-based multiplicative updates in most particle filters that are known to precipitate the numerical problem of weight collapse within a finite-ensemble setting. A study to estimate the a-priori error bounds in the proposed scheme is undertaken. The numerical evidence, presently gathered from the assessed performance of the proposed and a few other competing filters on a class of nonlinear dynamic system identification and target tracking problems, is suggestive of the remarkably improved convergence and accuracy of the new filter.

Benchmark approach for defaultable claims under partial information

We study the pricing and hedging of defaultable claims under the benchmark approach, introduced by Platen and Heath (2006), with partial information. The model is considered to be driven by an Ornstein–Uhlenbeck (OU) process. We fist convert the partial information model to a complete information model through the innovation process approach by deriving the non-linear filtering equation for the unobservable processes. We then define the total cash flow and use the Schweizer (2008) methodology for hedging the cash flow under the real world probability measure.

On absolutely continuous compensators and nonlinear filtering equations in default risk models

We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.

Stability of nonlinear filters in nonmixing case

The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure.

Optimal speed of detection in generalized Wiener disorder problems

We define a general Wiener disorder problem in which a sudden change in a time profile of unknown size has to be detected in white noise of small intensity. Since both the time of the change and its size are unknown, this problem is considerably harder than standard Wiener disorder problems where the size of the change is assumed to be known a priori. We formulate the problem within the Bayesian framework of nonlinear filtering theory, and use Strassen's functional law of the iterated logarithm to bound stochastic measures which arise in the nonlinear filtering equations. This leads to explicit expressions for the detection delay in the optimal statistics for small noise intensities, and we indicate how these can be used to analyse the detection delays of recursive suboptimal detection algorithms for this problem.

Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering

In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The Glivenko–Cantelli and Donsker theorems presented in this work extend the corresponding statements in the classical theory and apply to a class of genetic type particle numerical schemes of the nonlinear filtering equation.

Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities

This paper introduces in detail a new systematic method to construct approximate finite-dimensional solutions for the nonlinear filtering problem. Once a finite-dimensional family is selected, the nonlinear filtering equation is projected in Fisher metric on the corresponding manifold of densities, yielding the projection filter for the chosen family. The general definition of the projection filter is given, and its structure is explored in detail for exponential families. Particular exponential families which optimize the correction step in the case of discrete-time observations are given, and an a posteriori estimate of the local error resulting from the projection is defined. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented. The classical concept of assumed density filter (ADF) is compared with the projection filter. It is shown that the concept of ADF is inconsistent in the sense that the resulting filters depend on the choice of a stochastic calculus, i.e. the Ito or the Stratonovich calculus. It is shown that in the context of exponential families, the projection filter coincides with the Stratonovich-based ADF. An example is provided, which shows that this does not hold in general, for non-exponential families of densities.

Computational Issues In a Nonlinear Observer for Systems With Quantized Outputs

In this paper we develop an observer for nonlinear systems with quantized outputs. The observer is a recursive algorithm based on the intersection of sets: each measurement defines a set in state space which, by recursive intersection, is used to refine knowledge of the state. We develop the necessary data structures and procedures to implement the algorithm numerically. Comparisons are drawn between the proposed observer, the Kalman filter, and the equations of nonlinear filtering. Estimates are given for the error due to the triangulation of the set of consistent states and the computational complexity of the numerical implementation of our observer. Finally, the algorithm is applied to two example systems.

On a New Approach to the Solution of the Nonlinear Filtering Equation of Jump Processes

An implementable on-line approach to solve the nonlinear filtering equation for a partially observed system in which both the state and observation processes are jump processes is presented. By making use of the special structure of jump processes, the new method allows us to obtain the solution of the resulting nonlinear filtering equation from solving a linear system of ordinary differential equations and a linear system of algebraic equations recursively and via a simple normalization procedure.

Pathwise approximation and simulation for the Zakai filtering equation through operator splitting

Pathwise approximate solutions to the Zakai nonlinear filtering equation are analyzed and computed through operator splitting techniques. Several examples are presented to test the performances of the operator splitting algorithms.

Quantum-mechanical representations of nonlinear filtering and stochastic optimal control

The quantum-mechanical representation of nonlinear filtering and stochastic optimal control is investigated. By introducing wave functions which give the connections between the nonlinear filtering equation and the dynamic programming equation, two versions of quantum-mechanical wave equations called the Schrödinger-Langevin and the Itô-Schrödinger equations are derived. Furthermore, the probabilistic interpretation of the wave functions is discussed.

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitableL2 space. It is assumed that the Markov semigroup corresponding to the state process isC0 on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.

On the Reference Probability Approach to the Equations of Non-Linear Filtering

The filtering of diffusions from their noisy observations is considered in this paper. The introduction of various reference probability measures and the use of a stochastic Feynman-Kac formula is shown to lead to new and already known filtering equations. In some cases, which include extensions of the Benes filtering problem, the new equations we propose possess a nice Gaussian solution, yielding an explicit finite dimensional filter

On the Relation of Zakai’s and Mortensen’s Equations

The problem of optimal control for partially observable diffusion processes is studied by the dynamic programming in function space approach first proposed by R. E. Mortensen. The density of the conditional distribution of the (unobservable) signal given past and present observations, which satisfies the Zakai equation of nonlinear filtering, is viewed as the new state of the system. A verification result is established for the corresponding Bellman–Hamilton–Jacobi equation, known as Mortensen’s equation.

Solution of certain parabolic equations with unbounded coefficients and its application to nonlinear filtering

A probabilistic representation for the solution of a parabolic partial differential equation with uniformly elliptic diffusion coefficients but unbounded drift is obtained and some growth estimates are calculated. These are applied to prove uniqueness of solutions of a certain class of nonlinear filtering equations