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.

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