Stochastic Filtering for Diffusion Processes With Level Crossings

• Published in: IEEE transactions on automatic control Vol. 56; no. 9; pp. 2201 - 2206
• Main Authors: Capponi, A., Fatkullin, I., Ling Shi
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-09-2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Approximation methods; Automatic control; Computer science; control theory; systems; Computer simulation; Control theory. Systems; Density; Diffusion; Diffusion processes; Distribution theory; Eigenvalues and eigenfunctions; Equations; Exact sciences and technology; Filtering; Filtration; Fokker-Planck equation; Level crossings; Markov processes; Mathematical model; Mathematics; Modelling and identification; Monte Carlo methods; nonlinear filtering; Probability and statistics; Probability density function; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Thresholds; Level crossing; Density of states; Statistical distribution; Diffusion processes; Non linear filter; Fokker-Planck equation; Stochastic process; Gaussian process; Fokker Planck equation; Observer; Series expansion; Diffusion process; Filtering theory; System identification; nonlinear filtering; Threshold function; Numerical convergence
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2011.2157404

We provide a general framework for computing the state density of a noisy system given the sequence of hitting times of predefined thresholds. Our method relies on eigenfunction expansion corresponding to the Fokker-Planck operator of the diffusion process. For illustration, we present a particular example in which the state and the noise are one-dimensional Gaussian processes and observations are generated when the magnitude of the observed signal is a multiple of some threshold value. We present numerical simulations confirming the convergence and the accuracy of the recovered density estimator. Applications of the filtering methodology will be illustrated.