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In many areas of human endeavour, the systems involved are not available for direct measurement. Instead, by combining mathematical models for a system's evolution with partial observations of its evolving state, we can make reasonable inferences about it. The increasing complexity of the modern world makes this analysis and synthesis of high-volume data an essential feature in many real-world problems.

The celebrated Kalman-Bucy filter, designed for linear dynamical systems with linearly structured measurements, is the most famous Bayesian filter. Its generalizations to nonlinear systems and/or observations are collectively referred to as nonlinear filtering (NLF), an extension of the Bayesian framework to the estimation, prediction, and interpolation of nonlinear stochastic dynamics. NLF uses a stochastic model to make inferences about an evolving system and is a theoretically optimal algorithm.

The breadth of its applications, firmly established and still emerging, is simply astounding. Early uses such as cryptography, tracking, and guidance were mostly of a military nature. Since then, the scope has exploded. It includes the study of global climate, estimating the state of the economy, identifying tumours using non-invasive methods, and much more.

The Oxford Handbook of Nonlinear Filtering is the first comprehensive written resource for the subject. It contains classical and recent results and applications, with contributions from 58 authors. Collated into 10 parts, it covers the foundations of nonlinear filtering, connections to stochastic partial differential equations, stability and asymptotic analysis, estimation and control, approximation theory and numerical methods for solving the nonlinear filtering problem (including particle methods). It also contains a part dedicated to the application of nonlinear filtering to several problems in mathematical finance.

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In many areas of human endeavour, the systems involved are not available for direct measurement. Instead, by combining mathematical models for a system's evolution with partial observations of its evolving state, we can make reasonable inferences about it. The increasing complexity of the modern world makes this analysis and synthesis of high-volume data an essential feature in many real-world problems.

The celebrated Kalman-Bucy filter, designed for linear dynamical systems with linearly structured measurements, is the most famous Bayesian filter. Its generalizations to nonlinear systems and/or observations are collectively referred to as nonlinear filtering (NLF), an extension of the Bayesian framework to the estimation, prediction, and interpolation of nonlinear stochastic dynamics. NLF uses a stochastic model to make inferences about an evolving system and is a theoretically optimal algorithm.

The breadth of its applications, firmly established and still emerging, is simply astounding. Early uses such as cryptography, tracking, and guidance were mostly of a military nature. Since then, the scope has exploded. It includes the study of global climate, estimating the state of the economy, identifying tumours using non-invasive methods, and much more.

The Oxford Handbook of Nonlinear Filtering is the first comprehensive written resource for the subject. It contains classical and recent results and applications, with contributions from 58 authors. Collated into 10 parts, it covers the foundations of nonlinear filtering, connections to stochastic partial differential equations, stability and asymptotic analysis, estimation and control, approximation theory and numerical methods for solving the nonlinear filtering problem (including particle methods). It also contains a part dedicated to the application of nonlinear filtering to several problems in mathematical finance.

Features

  • Comprehensive, providing a unique reference source for all areas of nonlinear (or stochastic) filtering
  • Up-to-date, giving a modern interpretation of the classical theory, with many recent results
  • Covers theory and practice, including modern applications of nonlinear filtering, or hidden Markov models (HMM)
  • Authoritative, with expert editors and 58 contributors who are leaders in the field

1Introduction
2The Foundations of Nonlinear Filtering
2.1Nonlinear Filtering Problems I. Bayes Formulae and Innovations
2.2Nonlinear Filtering Problems II. Associated Equations
2.3Nonlinear Filtering Equations for Processes With Jumps
2.4The Filtered Martingale Problem
3Nonlinear Filtering and Stochastic Partial Differential Equations
3.1Filtering Equations for Partially Observable Diffusion Processes With Lipschitz Continuous Coefficients
3.2Malliavin Calculus Applications to the Study of Nonlinear Filtering
3.3Chaos Expansion to Nonlinear Filtering
4Stability and Asymptotic Analysis
4.1On Filtering with Unspecified Initial Data for Non-uniformly Ergodic Signals
4.2Exponential Decay Rate of the Filter's Dependence on the Initial Distribution
4.3Intrinsic Methods in Filter Stability
4.4Feller and Stability Properties of the Nonlinear Filter
4.5Lipschitz Continuity of Feynman-Kac Propagators
5Special Topics
5.1Pathwise Nonlinear Filtering
5.2The Innovation Problem
5.3Nonlinear Filtering and Fractional Brownian Motion
6Estimation and Control
6.1Dual Filters, Path Estimators and Information
6.2Filtering for Discrete-Time Markov Processes and Applications to Inventory Control with Incomplete Information
6.3Bayesian Filtering of Stochastic Hybrid Systems in Discrete-time and Interacting Multiple Model
7Approximation Theory
7.1Error Bounds for the Nonlinear Filtering of Diffusion Processes
7.2Discretizing the Continuous Time Filtering Problem. Order of Convergence
7.3Large Sample Asymptotics for the Ensemble Kalman Filter
8The Particle Approach
8.1Particle Approximations to the Filtering Problem in Continuous Time
8.2Tutorial on Particle Filtering and Smoothing: Fifteen Years Later
8.3A Mean Field Theory of Nonlinear Filtering
8.4The Particle Filter in Practice
8.5Introducing Cubature to Filtering
9Numerical Methods in Nonlinear Filtering
9.1Numerical Approximations to Optimal Nonlinear Filters
9.2Signal Processing Problems on Function Space: Bayesian Formulation, SPDEs and Effective MCMC Methods
9.3Robust, Computationally Efficient Algorithms for Tracking Problems with Measurement Process Nonlinearities
9.4Nonlinear Filtering Algorithms Based on Averaging Over Characteristics and on the Innovation Approach
10Nonlinear Filtering in Financial Mathematics
10.1Nonlinear Filtering in Models for Interest-Rate and Credit Risk
10.2An Asset Pricing Model with Mean Reversion and Regime Switching Stochastic Volatility
10.3Portfolio Optimization Under Partial Observation: Theoretical and Numerical Aspects
10.4Filtering with Counting Process Observations: Application to the Statistical Analysis of the Micromovement of Asset Price
Aimed mainly at graduate level and above, the Handbook will appeal to both pure and applied mathematicians, statisticians, economists, engineers and other scientists for whom filtering theory is important.
  • The Oxford Handbook of Nonlinear Filtering



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