Asymptotic Analysis for Piecewise Linear Filtering

A discrete-time nonlinear filtering problem with piecewise linear coefficients and not necessarily Gaussian disturbances is considered. It is shown that it possesses asymptotic properties that coincide with the analogous properties of a filtering problem for a suitably randomized linear model which admits a finite-dimensional solution. The asymptotic properties are connected to the behavior of the nonlinear filters when some parameters of the distribution of the initial condition and of the signal disturbances become small. These asymptotic properties allow to consider the finite-dimensional filter as an approximate solution to the original problem. It can in fact be shown that, asymptotically, the original and the approximate models have the same conditional moments and, in particular, the same conditional mean square errors

Small Noise Analysis for Piecewise Linear Stochastic Control Problems

A discrete-time stochastic control problem is considered for a dynamical model with piecewise linear coefficients and not necessarily Gaussian disturbances. The cost criteria and the class of admissible controls include piecewise polynomial costs and piecewise linear controls respectively. It is shown that relevant asymptotic (for vanishing noise) properties of this problem coincide with the corresponding properties of a suitably chosen adaptive control problem with linear dynamics. In particular, it turns out that the optimal values of the two problems tend to coincide and that almost optimal controls for one problem are almost optimal also for the other

On Observability of Chaotic Systems: An Example

The concept of observability of a special chaotic system, namely the dyadic map, is studied here in case the observation is not exact. The usual concept of observable subspace does not distinguish among the behavior of different models. It turns out that a suitable extension of this concept can be obtained using the idea of Hausdorff dimension. It is shown that this dimension increases as the observation error becomes smaller, and is equal to one only if the system is observable

Hermite Polynomials Expansions for Discrete-Time Nonlinear Filtering

A finite-dimensional approximation to general discrete-time nonlinear filtering problems is provided. It consists in a direct approximation to the recursive Bayes formula, based on a Hermite polynomials expansion of the transition density of the signal process. The approximation is in the sense of convergence, in a suitable weighted norm, to the conditional density of the signal process given the observations. The choice of the norm is in turn made so as to guarantee also the convergence of the conditional moments as well as to allow the evaluation of an upper bound for the approximation error

Almost sure optimality and optimality in probability for stochastic control problems over an infinite time horizon

A pathwise optimality criterion is proposed for stochastic control problems in order toreduce the risk connected with the fluctuations of the cost around its expected value. Thisapproach may be of relevance also in economic applications, where risky situations appearparticularly dangerous. Some examples of applications are examined, in particular for thelinear quadratic Gaussian model

Pathwise optimality in stochastic control

We introduce a notion of pathwise optimality for stochastic control problems over an infinite time horizon, and give sufficient conditions for the existence of pathwise optimal controls. We analyze both diffusion processes and processes with discrete state space