Nonlinear filtering with correlated Lévy noise characterized by copulas

Abstract

The objective in stochastic filtering is to reconstruct the information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process.Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Lévy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Lévy processes. More precisely, L = (L1, L2) is a 2-dimensional Lévy process in which the structure of dependence is described by a Lévy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and L for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P(X(t) > a), where a is a threshold.Primary 60H15, 35R30, 60K35; secondary 46G10, 28B05

A Particle Filter for Nonlinear Filtering With L\'evy Jumps

“…Let us denote the distribution of the initial data x 0 by χ. Let us assume that the state process X is a solution to (23) and the continuous part of the observation process Y driven by ( 24) is given at grid points…”

“…This possibility of considering different tail dependences was the motivation to take the Gumbel copula. For details on how to generate random variable conditioned by a copula, we refer to the appendix A and to the books [9,60,23].…”

“…If G is invertible, generate a on [0, 1] uniform distributed random variable U 2 and put Y := G −1 U 2 (U 1 ). For further details on simulating copulas, we refer to [23].…”

“…In this way, a copula can be seen as a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform. For an introduction to copulas we refer to the books [9,60,23].…”

“…In a second step, we deal with the Gaussian noise, by implementing incorporating the observation by adding some weights. That means we To model the dependence structure of the different Lévy processes, we use the concepts of copulas, already used in [23].…”