Asymptotic error distributions of the Euler method for continuous-time nonlinear filtering

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

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A Kusuoka–Lyons–Victoir particle filter

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0076 ◽

2013 ◽

Vol 469 (2156) ◽

pp. 20130076 ◽

Cited By ~ 3

Author(s):

Dan Crisan ◽

Salvador Ortiz-Latorre

Keyword(s):

Particle Filter ◽

Continuous Time ◽

Nonlinear Filtering ◽

Computational Effort ◽

Observation Data ◽

Minimal Variance ◽

Discretization Grid ◽

Filtering Problem ◽

Grid Mesh ◽

Number Of Particles

The aim of this paper is to introduce a new numerical algorithm for solving the continuous time nonlinear filtering problem. In particular, we present a particle filter that combines the Kusuoka–Lyons–Victoir (KLV) cubature method on Wiener space to approximate the law of the signal with a minimal variance ‘thinning’ method, called the tree-based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of our approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle filter in terms of the computational effort (number of particles) and the discretization grid mesh. Finally, we test the performance of the new algorithm on a benchmark problem (the Beneš filter).

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Information geometric nonlinear filtering

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025715500149 ◽

2015 ◽

Vol 18 (02) ◽

pp. 1550014 ◽

Cited By ~ 7

Author(s):

Nigel J. Newton

Keyword(s):

Continuous Time ◽

Nonlinear Filtering ◽

Riemannian Metric ◽

Point Of View ◽

Posterior Distributions ◽

Hilbert Manifold ◽

Information Theoretic ◽

Finite Dimensional ◽

Geometric Representations ◽

Filtering Problem

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's -1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.

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The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

Journal of Applied Probability ◽

10.1239/jap/1429282612 ◽

2015 ◽

Vol 52 (1) ◽

pp. 149-166 ◽

Cited By ~ 2

Author(s):

Hanchao Wang

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Stochastic Differential Equation ◽

Stochastic Differential Equations ◽

Lévy Processes ◽

Levy Processes ◽

Euler Scheme ◽

Asymptotic Error ◽

Error Distributions ◽

Itô Semimartingale

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

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Nonlinear filtering schemes for continuous-time stochastic hybrid systems

Control Engineering Practice ◽

10.1016/0967-0661(95)90205-8 ◽

1995 ◽

Vol 3 (1) ◽

pp. 127

Keyword(s):

Hybrid Systems ◽

Continuous Time ◽

Nonlinear Filtering ◽

Stochastic Hybrid Systems

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Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4037780 ◽

2017 ◽

Vol 140 (3) ◽

Cited By ~ 8

Author(s):

Amirhossein Taghvaei ◽

Jana de Wiljes ◽

Prashant G. Mehta ◽

Sebastian Reich

Keyword(s):

Continuous Time ◽

Nonlinear Filtering ◽

Sequential Monte Carlo ◽

Optimal Transport ◽

Nonlinear Problems ◽

Improved Stability ◽

Consistent Solution ◽

Gaussian Case ◽

Non Gaussian ◽

Filtering Problem

This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman–Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman–Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman–Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

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