3,085 research outputs found
An Elementary Approach to Filtering in Systems with Fractional Brownian Observation Noise
The problem of optimal filtering is addressed for a signal observed through a possibly nonlinear channel driven by a fractional Brownian motion. An elementary and completely self-contained approach is developed. An appropriate Girsanov type result is proved and a process -- equivalent to the innovation process in the usual situation where the observation noise is a Brownian motion -- is introduced. Zakai's approach is partly extended to derive filtering equations when the signal process is a diffusion. The case of conditionally Gaussian linear systems is analyzed. Closed form equations are derived both for the mean of the optimal filter and the conditional variance of the filtering error. The results are explicit in various special cases
Fractional generalizations of filtering problems and their associated fractional Zakai equations
In this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process
Linear filtering of systems with memory
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.Comment: Full names are use
Statistical analysis of the mixed fractional Ornstein--Uhlenbeck process
This paper addresses the problem of estimating drift parameter of the
Ornstein - Uhlenbeck type process, driven by the sum of independent standard
and fractional Brownian motions. The maximum likelihood estimator is shown to
be consistent and asymptotically normal in the large-sample limit, using some
recent results on the canonical representation and spectral structure of mixed
processes.Comment: to appear in Theory of Probability and its Application
- …