The Oxford Handbook of Nonlinear Filtering

Hardcover | March 3, 2011

EditorDan Crisan, Boris Rozovskii

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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 themodern 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 Bayesianframework 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 stochasticpartial 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 severalproblems 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 themodern world makes this analysis and synthesis o...

Dan Crisan is Reader in Mathematics at Imperial College London. His main research interest is stochastic filtering theory. Boris Rozovskii is Ford Foundation Professor at Brown University. His main interests are in stochastic partial differential equations (SPDEs) and their applications.

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Fundamentals of Stochastic Filtering
Fundamentals of Stochastic Filtering

Hardcover|Oct 23 2008


Format:HardcoverDimensions:1088 pages, 9.69 × 6.73 × 0.1 inPublished:March 3, 2011Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199532907

ISBN - 13:9780199532902


Extra Content

Table of Contents

1. D. Crisan, B. Rozovsky: Introduction2. The Foundations of Nonlinear FilteringH. Kunita: 2.1 Nonlinear Filtering Problems I. Bayes Formulae and InnovationsH. Kunita: 2.2 Nonlinear Filtering Problems II. Associated EquationsB. Grigelionis and R. Mikulevicius: 2.3 Nonlinear Filtering Equations for Processes With JumpsT. G. Kurtz and G. Nappo: 2.4 The Filtered Martingale Problem3. Nonlinear Filtering and Stochastic Partial Differential EquationsN. V. Krylov: 3.1 Filtering Equations for Partially Observable Diffusion Processes With Lipschitz Continuous CoefficientsM. Chaleyat-Maurel: 3.2 Malliavin Calculus Applications to the Study of Nonlinear FilteringS. V. Lototsky: 3.3 Chaos Expansion to Nonlinear Filtering4. Stability and Asymptotic AnalysisM. L. Kleptsyna and A. Y. Veretennikov: 4.1 On Filtering with Unspecified Initial Data for Non-uniformly Ergodic SignalsR. Atar: 4.2 Exponential Decay Rate of the Filter's Dependence on the Initial DistributionP. Chigansky, R. Liptser and R. Van Handel: 4.3 Intrinsic Methods in Filter StabilityA. Budhiraja: 4.4 Feller and Stability Properties of the Nonlinear FilterW. Stannat: 4.5 Lipschitz Continuity of Feynman-Kac Propagators5. Special TopicsM. Davis: 5.1 Pathwise Nonlinear FilteringA. J. Heunis: 5.2 The Innovation ProblemT. Duncan: 5.3 Nonlinear Filtering and Fractional Brownian Motion6. Estimation and ControlN. J. Newton: 6.1 Dual Filters, Path Estimators and InformationA. Bensoussan, M. Cakanyildirim and S. P. Sethi: 6.2 Filtering for Discrete-Time Markov Processes and Applications to Inventory Control with Incomplete InformationH. A.P. Blom and Y. Bar-Shalom: 6.3 Bayesian Filtering of Stochastic Hybrid Systems in Discrete-time and Interacting Multiple Model7. Approximation TheoryO. Zeitouni: 7.1 Error Bounds for the Nonlinear Filtering of Diffusion ProcessesD. Crisan: 7.2 Discretizing the Continuous Time Filtering Problem. Order of ConvergenceF. Le Gland, V. Monbet and V.-D. Tran: 7.3 Large Sample Asymptotics for the Ensemble Kalman Filter8. The Particle ApproachJ. Xiong: 8.1 Particle Approximations to the Filtering Problem in Continuous TimeA. Doucet and A. M. Johansen: 8.2 Tutorial on Particle Filtering and Smoothing: Fifteen Years LaterP. Del Moral, F. Patras and S. Rubenthaler: 8.3 A Mean Field Theory of Nonlinear FilteringT. B. Schon, F. Gustafsson and R. Karlsson: 8.4 The Particle Filter in PracticeC. Litterer and T. Lyons: 8.5 Introducing Cubature to Filtering9. Numerical Methods in Nonlinear FilteringH. J. Kushner: 9.1 Numerical Approximations to Optimal Nonlinear FiltersM. Hairer, A. Stuart and J. Voss: 9.2 Signal Processing Problems on Function Space: Bayesian Formulation, SPDEs and Effective MCMC MethodsJ. M. C. Clark and R. B. Vinter: 9.3 Robust, Computationally Efficient Algorithms for Tracking Problems with Measurement Process NonlinearitiesG.N. Milstein and M. Tretyakov: 9.4 Nonlinear Filtering Algorithms Based on Averaging Over Characteristics and on the Innovation Approach10. Nonlinear Filtering in Financial MathematicsR. Frey and W. Runggaldier: 10.1 Nonlinear Filtering in Models for Interest-Rate and Credit RiskR. J. Elliott, H. Miao and Z. Wu: 10.2 An Asset Pricing Model with Mean Reversion and Regime Switching Stochastic VolatilityH. Pham: 10.3 Portfolio Optimization Under Partial Observation: Theoretical and Numerical AspectsL. C. Scott and Y. Zeng: 10.4 Filtering with Counting Process Observations: Application to the Statistical Analysis of the Micromovement of Asset Price