Asymptotic stability of the optimal filter for random chaotic maps

Abstract

The asymptotic stability of the optimal filtering process in discrete time is revisited. The filtering process is the conditional probability of the state of a Markov process, called the signal process, given a series of observations. Asymptotic stability means that the distance between the true filtering process and a wrongly initialised filter converges to zero as time progresses. In the present setting, the signal process arises through iterating an i.i.d. sequence of uniformly expanding random maps. It is showed that for such a signal, the asymptotic stability is exponential provided that its initial conditions are sufficiently smooth. Similar to previous work on this problem, Hilbert's projective metric on cones is employed as well as certain mixing properties of the signal, albeit with important differences. Mixing and ultimately filter stability in the present situation are due to the expanding dynamics rather than the stochasticity of the signal process. In fact, the conditions even permit iterations of a fixed (nonrandom) expanding map.

Exponential stability and asymptotic properties of the optimal filter for signals with deterministic hyperbolic dynamics

“…It is easy to see that if the distribution π 0 of X 0 is invariant and ergodic with respect to K, the entire signal process is ergodic and, by standard arguments, is indeed defined also for negative times. The observations can be likewise extended to negative times, and a minor modification of the proof of Lemma 2.5 in [11] will show that the joint signal-observation process {(X k , Y k ); k ∈ Z} is a stationary and ergodic process.…”

“…In this section, we clarify the effect of applying the operator L ω to elements of A , C , and of D. In fact, in case of the cone D, an operator which in a sense is the dual of L will have to be considered. By Equation ( 13) and (11) we have…”

“…Again due to the separability of R d , we can assume that g is measurable on (R d × M, B d ⊗ B M ) (see [11], item (3) of Lemma A.1.). Further, due to Tonelli's theorem we also have that for any probability measure ν on (M, B M ),…”

“…In [11] exponential stability of the filtering process is demonstrated for signals produced by random expanding maps, provided that the initial condition of the filter is sufficiently smooth. The results rely on the dynamical properties of expanding maps, rather than on the stochasticity and in fact include the case of deterministic expanding maps.…”

“…In contrast to previous works using the Hilbert metric on cones though, due to the dependence on the (random) observation, the operator Lω will not be a contraction under the Hilbert metric on a single cone. Rather, as in [11], we need to construct a random cone C ω which is invariant under Lω in the sense that Lω C ω ⊂ C T ω , and so that the Hilbert projective metric is contracted. The regular probability kernel µ referred to in our main result will then emerge as a kind of random (or pullback) fixed point of Lω .…”