We study the linear filtering problem for systems driven by continuous Gaussian processes with memory described by two parameters. The driving processes have the virtue that they possess stationary increments and simple semimartingale representations simultaneously. It allows for straightforward parameter estimations. After giving the semimartingale representations of the processes by innovation theory, we derive Kalman-Bucy-type filtering equations for the systems. We apply the result to the optimal portfolio problem for an investor with partial observations. We illustrate the tractability of the filtering algorithm by numerical implementations.
The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a new time discretisation of these functionals corresponding to a chosen partition of the time interval and show that the convergence rate of discretisation is proportional with the square of the mesh of the partition.
The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the $m$-power of the mesh of the partition for arbitrary $m\in\math…
Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.
An algorithm for the estimation of multiple targets from partial and corrupted observations is introduced based on the concept of partially-distinguishable multi-target system. It combines the advantages of engineering solutions like MHT with the rigour of point-process-based methods. It is demonstrated that under intuitive assumptions and approximations, the complexity of the proposed multi-target estimation algorithm can be made linear in the number of tracks and linear in the number of observations, while naturally preserving distinct tracks for detected targets, unlike point-process-based …
We give meaning to differential equations with a rough path term and a Brownian noise term as driving signals. Such differential equations as well as the question of regularity of the solution map arise naturally and we discuss two applications: one revisits Clark's robustness problem in nonlinear filtering, the other is a Feynman--Kac type representation of linear RPDEs. En passant, we give a short and direct argument that implies integrability estimates for rough differential equations with Gaussian driving signals which is of independent interest.
Motivated by the simulation of stable random fields, we consider the issue of discrete approximations of independently scattered stable noise. Two approaches are proposed: grid approximations available when the underlying space is $\bbR^d$ and shot noise approximations available on more general spaces. Limit theorems stating the convergence of discrete random noises to stable white noise are proved. These results are then applied to study moving average spatial random fields with heavy-tailed innovations and related limit theorems. A second application deals with discrete approximation for Bro…
We consider an affine process $X$ which is only observed up to an additive white noise, and we ask for its law, for some time $t > 0 $, conditional on all observations up to this time $ t $. This is a general, possibly high dimensional filtering problem which is not even locally approximately Gaussian, whence essentially only particle filtering methods remain as solution techniques. In this work we present an efficient numerical solution by introducing an approximate filter for which conditional characteristic functions can be calculated by solving a system of generalized Riccati differential …
Filtering problems with general exponential quadratic criteria are investigated for Gauss-Markov processes. In this setting, the Linear Exponential Gaussian and Risk-Sensitive filtering problems are solved and it is shown that they may have different solutions.
Filtering is a widely used methodology for the incorporation of observed data into time-evolving systems. It provides an online approach to state estimation inverse problems when data is acquired sequentially. The Kalman filter plays a central role in many applications because it is exact for linear systems subject to Gaussian noise, and because it forms the basis for many approximate filters which are used in high dimensional systems. The aim of this paper is to study the effect of model error on the Kalman filter, in the context of linear wave propagation problems. A consistency result is pr…
A direct approach to linear backward filtering equations for SDE systems is proposed. This preprint is a corrected version of the paper 1995 in the LMS Lecture Notes combined with another paper by the author on the direct approach to linear SPDEs for SDEs.
The classical state-space approach to optimal estimation of stochastic processes is efficient when the driving noises are generated by martingales. In particular, the weight function of the optimal linear filter, which solves a complicated operator equation in general, simplifies to the Riccati ordinary differential equation in the martingale case. This reduction lies in the foundations of the Kalman-Bucy approach to linear optimal filtering. In this paper we consider a basic Kalman-Bucy model with noises, generated by independent fractional Brownian motions, and develop a new method of asympt…
We consider high order approximations of the solution of the stochastic filtering problem, derive their pathwise representation in the spirit of the earlier work of Clark and Davis and prove their robustness property. In particular, we show that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space. This property is important from the practical point of view as it is in fact the pathwise version of the filtering functional that is sought in numerical applications. Moreover, the pathwise viewpoint will be a st…
This paper revisits the question of duality between minimum variance estimation and optimal control first described for the linear Gaussian case in the celebrated paper of Kalman and Bucy. A duality result is established for nonlinear filtering, mirroring closely the original Kalman-Bucy duality of control and estimation for linear systems. The result for the finite state-space continuous time Markov chain is presented. It's solution is used to derive the classical Wonham filter.
It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories, allowing for a convenient handling of state constraints and meeting points. We now extend this viewpoint to estimation problems where it is known that kernels are the covariances of stochastic processes. Here, the Markovian Gaussian processes stem from the linear stochastic differential equations describing the continuous-time dynamics and observations. Taking…
In the following article we consider the numerical approximation of the non-linear filter in continuous-time, where the observations and signal follow diffusion processes. Given access to high-frequency, but discrete-time observations, we resort to a first order time discretization of the non-linear filter, followed by an Euler discretization of the signal dynamics. In order to approximate the associated discretized non-linear filter, one can use a particle filter (PF). Under assumptions, this can achieve a mean square error of $\mathcal{O}(ε^2)$, for $ε>0$ arbitrary, such that the associated …
Huazhong University of Science and Technology（华中科技大学）；Illinois Institute of Technology（伊利诺伊理工学院）
The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems with non-Gaussian Lévy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method.
