Filtering deals with recursive estimation of signals from their noisy measurements. A typical setup consists of a Markov process, which cannot be observed directly and is to be "filtered"from the trajectory of another process, related to it statistically. The general idea is to seek a "best estimate"for the true value of the signal, given only some (potentially noisy) observations of that signal. The optimal estimate is given by the conditional expectation and can be generated by a recursive equation, called the filtering equation, driven by the observation process. If the signal/observation model is linear and Gaussian, the filtering problem is called the Kalman-Bucy filter, otherwise is called a nonlinear filter. Being of considerable practical importance in engineering and economics, the filtering theory poses many interesting mathematical problems and it utilizes areas of mathematics such as stochastic calculus, martingales, etc. This thesis focuses on the mathematical aspects of nonlinear filtering for the case when the signal is a jump-diffusion process, i.e. a stochastic process that involves jumps and diffusion. One important objective of the thesis is to describe the evolution of the conditional distribution characterizing the optimal nonlinear filter using a stochastic differential equation known as the Zakai equation. The main contributions of the research are the moment estimates of the multi-dimensional jump-diffusion process which represent the signal in the nonlinear filtering problem, and a new approach for the uniqueness of the measure-valued solution of the stochastic differential equation that describes the evolution of the optimal filter. Applications of the nonlinear filtering theory to financial economics estimation problems including stochastic volatility models are discussed.

Nonlinear Filtering covers linear and nonlinear filtering in a comprehensive manner, with appropriate theoretic and practical development. Aspects of modeling, estimation, recursive filtering, linear filtering, and nonlinear filtering are presented with appropriate and sufficient mathematics. A modeling-control-system approach is used when applicable, and detailed practical applications are presented to elucidate the analysis and filtering concepts. MATLAB routines are included, and examples from a wide range of engineering applications - including aerospace, automated manufacturing, robotics, and advanced control systems - are referenced throughout the text.

Linear filtering techniques are used to develop a quasi maximum likelihood estimator for asymmetric stochastic volatility models. The estimator is straightforward to implement and performs well in Monte Carlo experiments.

This paper is concerned with filtering for various types of time series models including the class of generalized ARCH models and stochastic volatility models. We extend the results of Thavaneswaran and Abraham (1988) for some time series models using martingale estimating functions. Nonlinear filtering for biostatistical time series models with censored observations is also discussed as a special case.

The book combines both rigor and intuition to derive most of the classical results of linear and nonlinear filtering and beyond. Many fundamental results recently discovered by the author are included. Furthermore, many results that have appeared in recent years in the literature are also presented. The most interesting feature of the book is that all the derivations of the linear filter equations given in Chapters 3-11, beginning from the classical Kalman filter presented in Chapters 3 and 5, are based on one basic principle which is fully rigorous but also very intuitive and easily understandable. The second most interesting feature is that the book provides a rigorous theoretical basis for the numerical solution of nonlinear filter equations illustrated by multidimensional examples. The book also provides a strong foundation for theoretical understanding of the subject based on the theory of stochastic differential equations.

This book provides a comprehensive account of stochastic filtering as a modeling tool in finance and economics. It aims to present this very important tool with a view to making it more popular among researchers in the disciplines of finance and economics. It is not intended to give a complete mathematical treatment of different stochastic filtering approaches, but rather to describe them in simple terms and illustrate their application with real historical data for problems normally encountered in these disciplines. Beyond laying out the steps to be implemented, the steps are demonstrated in the context of different market segments. Although no prior knowledge in this area is required, the reader is expected to have knowledge of probability theory as well as a general mathematical aptitude. Its simple presentation of complex algorithms required to solve modeling problems in increasingly sophisticated financial markets makes this book particularly valuable as a reference for graduate students and researchers interested in the field. Furthermore, it analyses the model estimation results in the context of the market and contrasts these with contemporary research publications. It is also suitable for use as a text for graduate level courses on stochastic modeling.

NONLINEAR FILTERS Discover the utility of using deep learning and (deep) reinforcement learning in deriving filtering algorithms with this insightful and powerful new resource Nonlinear Filters: Theory and Applications delivers an insightful view on state and parameter estimation by merging ideas from control theory, statistical signal processing, and machine learning. Taking an algorithmic approach, the book covers both classic and machine learning-based filtering algorithms. Readers of Nonlinear Filters will greatly benefit from the wide spectrum of presented topics including stability, robustness, computability, and algorithmic sufficiency. Readers will also enjoy: Organization that allows the book to act as a stand-alone, self-contained reference A thorough exploration of the notion of observability, nonlinear observers, and the theory of optimal nonlinear filtering that bridges the gap between different science and engineering disciplines A profound account of Bayesian filters including Kalman filter and its variants as well as particle filter A rigorous derivation of the smooth variable structure filter as a predictor-corrector estimator formulated based on a stability theorem, used to confine the estimated states within a neighborhood of their true values A concise tutorial on deep learning and reinforcement learning A detailed presentation of the expectation maximization algorithm and its machine learning-based variants, used for joint state and parameter estimation Guidelines for constructing nonparametric Bayesian models from parametric ones Perfect for researchers, professors, and graduate students in engineering, computer science, applied mathematics, and artificial intelligence, Nonlinear Filters: Theory and Applications will also earn a place in the libraries of those studying or practicing in fields involving pandemic diseases, cybersecurity, information fusion, augmented reality, autonomous driving, urban traffic network, navigation and tracking, robotics, power systems, hybrid technologies, and finance.