Partially Observable Control Problems with Compulsory Shifts of the State

Stochastic control problems with partial state observation and the long-run average cost criterion are among the most difficult dynamic stochastic optimization problems and almost nothing has so far appeared in the literature concerning their solution. On the other hand many problems in Engineering, Operations Research, and the Economic and Social Sciences can be modelled as problems of the above type. In the present paper we study conditions under which the filtering process associated with the partially observed state process has a unique invariant measure and describe ways to approximate it. We finally discuss the applications of these results to the construction of nearly optimal controls

Combined Filtering and Parameter Estimation for Discrete-Time Systems Driven by Approximately White Gaussian Noise Disturbances

In the problem of combined filtering and parameter estimation one considers a stochastic dynamical system whose state x_t is only partially observed through an observation process y_t. The stochastic model for the process pair (x_t, y_t) depends furthermore on an unknown parameter theta. Given an observation history of the process y_t, the problem then consists in estimating recursively both the current state x_t of the system (filtering) as well as the value theta of the parameter (Bayesian parameter estimation). The problem is a rather difficult one: Even if, conditionally on a given value of theta, the process pair (x_t, y_t) satisfies a linear-Gaussian model so that the filtering problem for x_t can be solved via the familiar Kalman-Bucy filter; when theta is unknown, the problem becomes a difficult nonlinear filtering problem. The present paper, partly based on previous joint work of one of the authors, makes a contribution towards the solution of this problem in the case of discrete time and of a (conditionally on theta) linear model for x_t, y_t. The solution that is obtained is shown to be robust with respect to small variations in the a priori distributions in the model, in particular those of the disturbances

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

Arbitrage and utility maximization in market models with an insider

We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses an additional information in the form of an (Formula presented.)-measurable discrete random variable G, we give criteria for the no unbounded profits with bounded risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided. For the case when G is a continuous random variable, we consider the notion of no asymptotic arbitrage of the first kind (NAA1) and give an explicit construction for unbounded profits if NAA1 fails. © 2018 Springer-Verlag GmbH Germany, part of Springer Natur

A Bayesian adaptive control approach to risk management in a binomial model

We consider the problem of shortfall risk minimization when there is uncertainty about the exact stochastic dynamics of the underlying. Starting from the general discrete time model and the approach described in Runggaldier and Zaccaria (1999), we derive explicit analytic solutions for the particular case of a binomial model when there is uncertainty about the probability of an "up-movement". The solution turns out to be a rather intuitive extension of that for the classical Cox-Ross-Rubinstein model

Deterministic Approximation for Stochastic Control Problems

We consider a class of stochastic control problems where uncertainty is due to driving noises of general nature as well as to rapidly fluctuating processes affecting the drift. We show that, when the noise "intensity" is small and the fluctuations become fast, the stochastic problems can be approximated by a deterministic one. We also show that the optimal control of the deterministic problem is asymptotically optimal for the stochastic problems. Key Words : Stochastic and deterministic control, Stochastic differential equations, Weak convergence, Asymptotic optimality. AMS Subject Classification: 93E20, 93C15, 60B10, 60F17, 60G44, 49J15, 49K40, 49M45 Acknowledgement : This work was partially supported by GNAFA / CNR of the Italian National Research Council allowing a visit of the first and last authors at the University of Padova. The work of the last author was also supported by National Science Foundation Grant DMS 9301200 and NATO Scientific Exchange Grant CRG 900147 1.Introducti..