Related papers: The filtering equations revisited
Particle filters for data assimilation in nonlinear problems use "particles" (replicas of the underlying system) to generate a sequence of probability density functions (pdfs) through a Bayesian process. This can be expensive because a…
These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares…
We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement…
A novel approximate Bayesian filter based on backward stochastic differential equations is introduced. It uses a nonlinear Feynman--Kac representation of the filtering problem and the approximation of an unnormalized filtering density using…
The work concerns nonlinear filtering problems of stochastic differential equations with correlated L\'evy noises. First, we establish the Kushner-Stratonovich and Zakai equations through martingale representation theorems and the…
Filtering is concerned with the sequential estimation of the state, and uncertainties, of a Markovian system, given noisy observations. It is particularly difficult to achieve accurate filtering in complex dynamical systems, such as those…
Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We…
We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
This paper focuses on inverse problems to identify parameters by incorporating information from measurements. These generally ill-posed problems are formulated here in a probabilistic setting based on Bayes's theorem because it leads to a…
Computer simulations of differential equations require a time discretization, which inhibits to identify the exact solution with certainty. Probabilistic simulations take this into account via uncertainty quantification. The construction of…
We consider the problem of estimating the means $\mu_i$ of $n$ random variables $Y_i \sim N(\mu_i,1)$, $i=1,\ldots ,n$. Assuming some structure on the $\mu$ process, e.g., a state space model, one may use a summary statistics for the…
We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential…
We present, in the simplest possible form, the so called martingale problem strategy to establish limit theorems. The presentation is specially adapted to problems arising in partially hyperbolic dynamical systems. We will discuss a simple…
Nonstationary and nonlinear signals are ubiquitous in real life. Their decomposition and analysis is an important topic of research in signal processing. Recently a new technique, called Iterative Filtering, has been developed with the goal…
An important objective of the classical processing of stationary random sequences under nonparametric uncertainty is the problem of filtering in case when the distribution of the underlying signal is unknown. In this paper it is assumed…
State-space models can be used to incorporate subject knowledge on the underlying dynamics of a time series by the introduction of a latent Markov state-process. A user can specify the dynamics of this process together with how the state…
In this paper we consider the continuous--time nonlinear filtering problem, which has an infinite--dimensional solution in general, as proved by Chaleyat--Maurel and Michel. There are few examples of nonlinear systems for which the optimal…
This article proposes a new filtering model for stationary Gaussian Markov statistical experiments, given by diffusion-type difference stochastic equations.
The Markov modulated (switching) state space is an important model paradigm in applied statistics. In this article, we specifically consider Markov modulated nonlinear state-space models and address the online Bayesian inference problem for…