Lyapunov Exponents for Finite State Nonlinear Filtering

Abstract

The dependence of the optimal nonlinear filter on it's initial conditions is considered for continuous time linear filtering and for finite state space nonlinear filtering. Partial results are obtained in the high signal to noise ration case, together with a characterization of the Lyapunov exponent in the (easier) low SNR case.

An ergodic theorem for filtering with applications to stability

“…Indeed u m := log max |v|=1,v∈D U σ (τ (m − 1), τ m) • v are measurable with respect to F X (m−1)τ,mτ ∨ F B (m−1)τ,mτ and so stationarity and ergodicity are inherited from X, the increments of B and their independence. Integrability follows from the Gaussian properties of B and is verified similarly to Theorem 1.5 in [2].…”

“…Notably any solution of (3.6) started from a vector from S d−1 "picks up" the top Lyapunov exponent. This is a consequence of the Perron-Frobenious theorem applied in [2] to the positive stochastic flow generated by (3.6). The matrix ρ t ∧ρ t satisfies a linear SDE (see (3.23) below) as well and thus is in the scope of MET:…”

“…Unlike in (3.8), only inequality can be claimed in (3.9) since the flow ρ t ∧ρ t in not positive anymore and different initial conditions may in principle correspond to different Lyapunov exponents. Assembling (3.7), (3.8) and (3.9) together it is concluded in [2] that the stability of (1.2) is controlled by the Lyapunov spectral gap of (3.6):…”

“…Both (3.11) and (3.12) are obtained in [2] using the Feynman-Kac type formulae adapted to the filtering context.…”

“…Besides providing an additional insight into the problem, the method simplifies derivation of several already known upper bounds for γ ( [2], [6]). Also it allows to obtain a closed form formula and some higher order asymptotic expansions for γ in the two-dimensional case d = 2.…”