We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discrete-countable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...

this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise, and the classical "heat-bath" model of R. Kubo, modeling the motion of a "large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from 0. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel

. We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semiflow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate 1 (oe) of the trajectories expressed in terms of the noise variance oe. Roughly speaking we show that for small oe, 1 (oe) behaves like 0 oe 2 2 , while for large oe, it grows like log oe. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent 1 are provided. These estimates are sharp in the sense they reduce to known estimates in the deterministic or non-delay cases. 1. Introduction and Some Preliminaries. Lyapunov exponents for linear stochastic ordinary differential equations (without memory) have been studied by many authors; cf. for example, [AKO], [AOP], [B], Pardoux and Wihstutz[ PW1...

. We consider non-linear stochastic functional dierential equations (sfde's) on Euclidean space. We give sucient conditions for the sfde to admit locally compact smooth cocycles on the underlying innite-dimensional state space. Our construction is based on the theory of nite-dimensional stochastic ows and a non-linear variational technique. In Part II of this article, the above result will be used to prove a stable manifold theorem for non-linear sfde's. 1.

Abstract. The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove general existence and compactness theorems for C k-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinitedimensional noise. In Part 2, stationary solutions are viewed as cocycle-invariant random points in the infinite-dimensional state space. The pathwise local structure of solutions of semilinear see’s and spde’s near stationary solutions is described in terms of the almost sure longtime behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see’s and spde’s (Theorems 2.4.1-2.4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore, the local stable and unstable manifolds intersect transversally at the stationary point, and the unstable manifolds have fixed finite dimension. The proof uses infinite-dimensional multiplicative ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).

Stochastic filtering theory studies the problem of estimating an unobservable `signal' process X given the information obtained by observing an associated process Y (a `noisy' observation) within a certain time window [0; t]. It is possible to explicitly describe the distribution of X given Y in the setting of linear/gaussian systems. Outside the realm of the linear theory, it is known that only a few very exceptional examples have explicitly described posterior distributions. We present in detail a class of nonlinear filters (Benes filters) which allow explicit formulae. Using the explicit expression of the Laplace transform of a functional of Brownian motion we give a direct computation of the unnormalized conditional density of the signal for the Benes filter and obtain the formula for the normalized conditional density of X for two particular filters. In the case in which the signal X is a diffusion process and Y is given by the equation dY t = h(s; X s )ds+dW t ; where W is a Brownian moti...

Abstract not found