The Oxford Handbook of Nonlinear Filtering

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