Optimal Filtering of a Gaussian Signal in the Presence of Lévy Noise

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

“…We work in the onedimensional case using (4.17) and compare our results with those of [1] and [7]. The example we will look at will be that of infinite variance α-stable noisy obervations of a mean reverting Brownian motion, i.e.…”

“…Finally we take 100, 000 Monte-Carlo simulations to estimate the mean square error of the filter from the system, after which a median value of these were taken, the results are summarised in Table 1. Here "No filter" refers to the filter of this paper and LBM and AF are those developed in [7] and [1], respectively.…”

“…An exception to this is the paper [7], but that uses a very different approach to the one that we will present, and it seems that the only concrete examples that fit naturally into that framework are the α-stable Lévy processes. We also mention [1] that deals with a very similar problem to ours. We compare the approach of that paper with ours below.…”

“…In contrast to our linear filter, [1] obtain a non-linear optimal filter for the best measurable estimate. Having the system evolve with Gaussian noise is also a vital aspect of their set-up, indeed the authors describe this as a "significant limitation" in the closing paragraph of their paper.…”

“…In section 4 we describe the limiting processes, and obtain our main results. Some numerical simulations are presented in section 5, where we make practical comparisons of our filter with those of [1] and [7]. A more detailed account of the results presented here can be found in [6].…”