Nonlinear filtering of Itô-Lévy stochastic differential equations with continuous observations

Abstract

We study the n-dimensional nonlinear filtering problem for jumpdiffusion processes. The optimal filter is derived for the case when the observations are continuous. A proof of uniqueness is presented under fairly general circumstances.2000 Mathematics Subject Classification. Primary 60H15; Secondary 60J75.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

“…[19], Lemma 4.1) that under the reference measure P 0 , the process Z t is a standard Brownian motion independent of Y t , and Λ t satisfies the equation…”

“…Then, as is known, the optimal filtering solution of the filtering problem (3.3), (3.4) is given by the following Kallianpur-Striebel's formula [13,19,22])…”

“…Particular cases of this problem is discussed in Chapter 4 of [2] and in papers [19,20]. Namely, consider the following two cases:…”

“…However, many processes naturally arising in the modern science (in particular, in biology, genetics, finance) and engineering do not obey Gaussian driving processes. Filtering problems with the state and/or observation processes driven by Lévy processes were discussed in recent publications [2,19,20].…”

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

“…A more detailed account of the results presented here can be found in [6]. Finally we remark that non-linear filtering with Lévy processes has also recently received some development, and we direct the interested reader to [14,17,6]. …”