Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous - Time Non-Linear Systems

Summary. This article considers the application of particle filterin g to continuousdiscrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of state and noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.

[1]  Xiongzhi Chen Brownian Motion and Stochastic Calculus , 2008 .

[2]  P. Fearnhead,et al.  Particle filters for partially observed diffusions , 2007, 0710.4245.

[3]  Simo Särkkä,et al.  On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems , 2007, IEEE Trans. Autom. Control..

[4]  Dan Cornford,et al.  Gaussian Process Approximations of Stochastic Differential Equations , 2007, Gaussian Processes in Practice.

[5]  Jouko Lampinen,et al.  Rao-Blackwellized particle filter for multiple target tracking , 2007, Inf. Fusion.

[6]  Darren J. Wilkinson,et al.  Bayesian sequential inference for nonlinear multivariate diffusions , 2006, Stat. Comput..

[7]  B. Delyon,et al.  Simulation of conditioned diffusion and application to parameter estimation , 2006 .

[8]  P. Fearnhead,et al.  Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion) , 2006 .

[9]  Simo Särkkä,et al.  Recursive Bayesian inference on stochastic differential equations , 2006 .

[10]  Simo Sarkka On Sequential Monte Carlo Sampling of Discretely Observed Stochastic Differential Equations , 2006, 2006 IEEE Nonlinear Statistical Signal Processing Workshop.

[11]  M. Pitt,et al.  Likelihood based inference for diffusion driven models , 2004 .

[12]  D. Applebaum Lévy Processes and Stochastic Calculus: Preface , 2009 .

[13]  Rudolph van der Merwe,et al.  Sigma-point kalman filters for probabilistic inference in dynamic state-space models , 2004 .

[14]  D. Crisan Exact rates of convergeance for a branching particle approximation to the solution of the Zakai equation , 2003 .

[15]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .

[16]  E. L. Ionides,et al.  Inference and Filtering for Partially Observed Diusion Processes via Sequential Monte Carlo , 2003 .

[17]  A. Gallant,et al.  Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes , 2002 .

[18]  Geir Storvik,et al.  Particle filters for state-space models with the presence of unknown static parameters , 2002, IEEE Trans. Signal Process..

[19]  Thiagalingam Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation , 2001 .

[20]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice Using MATLAB , 2001 .

[21]  H. V. Trees Detection, Estimation, And Modulation Theory , 2001 .

[22]  Michael I. Jordan,et al.  Learning with Mixtures of Trees , 2001, J. Mach. Learn. Res..

[23]  Mohinder S. Grewal,et al.  Global Positioning Systems, Inertial Navigation, and Integration , 2000 .

[24]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[25]  Subhash Challa,et al.  Nonlinear filtering via generalized Edgeworth series and Gauss-Hermite quadrature , 2000, IEEE Trans. Signal Process..

[26]  P. Moral,et al.  Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering , 2000 .

[27]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[28]  D. Crisan,et al.  A particle approximation of the solution of the Kushner–Stratonovitch equation , 1999 .

[29]  Dan Crisan,et al.  Convergence of a Branching Particle Method to the Solution of the Zakai Equation , 1998, SIAM J. Appl. Math..

[30]  G. Kitagawa Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[31]  Robert F. Stengel,et al.  Optimal Control and Estimation , 1994 .

[32]  P. Kaye Infectious diseases of humans: Dynamics and control , 1993 .

[33]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[34]  B. Øksendal Stochastic differential equations : an introduction with applications , 1987 .

[35]  G. Kallianpur Stochastic filtering theory , 1979, Advances in Applied Probability.

[36]  A.H. Haddad,et al.  Applied optimal estimation , 1976, Proceedings of the IEEE.

[37]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[38]  Harry L. Van Trees,et al.  Detection, Estimation, and Modulation Theory, Part I , 1968 .

[39]  M. Alonso,et al.  Fundamental University Physics , 1967 .

[40]  A. Jazwinski Filtering for nonlinear dynamical systems , 1966 .

[41]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[42]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .