Preface

Stochastic control plays an important role in many scientific and applied disciplines. The goal of this book is to collect a group of outstanding investigations in various aspects of the stochastic systems and their behavior.

Linear discrete-time stochastic systems with uncertain observations (Chapter 1), and stochastic delay differential equations by noisy observations (Chapter 2) are considered at the beginning of the book. The model of coherent upper prevision, analyzed in Chapter 3, can be proposed to forecast in a chaotic system. A stochastic control problem where the system is governed by a nonlinear stochastic differential equation with jumps is explored in Chapter 4. The control is allowed to enter into both diffusion and jump terms.

A geometrical interpretation of different multi-agent systems evolving in phase space under the hypothesis of equiprobability is introduced, and some new results for statistical systems are obtained in Chapter 5. The process and procedures for stochastic modelling of structural elements are analyzed in Chapter 6.

The next three chapters are concerned with financial applications: a singular stochastic control model in European option hedging with transaction costs is studied in Chapter 7, and a stochastic optimal control problem for jump diffusions, where the controlled stochastic system is driven by both Brownian motion and Poisson random measure is investigated in Chapter 8. More precisely, the relationship between the maximum stochastic principle and the dynamic programming principle for the stochastic optimal control problem of jump diffusions is derived. Iterations for computing the maximal and stabilizing solution to the discrete time generalized algebraic Riccati equations are considered and their numerical properties are considered in Chapter 9.

The remaining chapters can be considered as a collection of several applications of the optimal control problems: a synthesis problem of the optimal control of the observation process based on a generalized probabilistic criteria (Chapter 10), and a problem for suggesting improvement of statistical decisions in revenue management systems under parametric uncertainty (Chapter 11). Next application is a problem for an acoustics optimization in the multimedia classroom. In Chapter 12, the stochastic based simulations and measurements of some objective parameters of acoustic quality

#### X Preface

and subjective evaluation of room acoustic quality are analyzed. A discrete-time stochastic epidemic model is introduced and examined from a statistical point of view in Chapter 13. The last studies the parameter identifiability of quantized linear systems with Gauss-Markov parameters from informational theoretic point of view.

> **Prof. Ivan Ganchev Ivanov** Head of the Department of Statistics and Econometrics, Faculty of Economics and Business Administration, Sofia University "St. Kl. Ohridski", Sofia, Bulgaria

X Preface

and subjective evaluation of room acoustic quality are analyzed. A discrete-time stochastic epidemic model is introduced and examined from a statistical point of view in Chapter 13. The last studies the parameter identifiability of quantized linear systems

**Prof. Ivan Ganchev Ivanov**

Bulgaria

Sofia University "St. Kl. Ohridski", Sofia,

Head of the Department of Statistics and Econometrics, Faculty of Economics and Business Administration,

with Gauss-Markov parameters from informational theoretic point of view.

**Chapter 0**

**Chapter 1**

**Design of Estimation Algorithms from an**

J. Linares-Pérez, R. Caballero-Águila and I. García-Garrido

and the initial state are Gaussian and mutually independent.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/45777

**1. Introduction**

**Innovation Approach in Linear Discrete-Time**

**Stochastic Systems with Uncertain Observations**

The least-squares estimation problem in linear discrete-time stochastic systems in which the signal to be estimated is always present in the observations has been widely treated; as is well known, the Kalman filter [12] provides the least-squares estimator when the additive noises

Nevertheless, in many real situations, usually the measurement device or the transmission mechanism can be subject to random failures, generating observations in which the state appears randomly or which may consist of noise only due, for example, to component or interconnection failures, intermittent failures in the observation mechanism, fading phenomena in propagation channels, accidental loss of some measurements or data inaccessibility at certain times. In these situations where it is possible that information concerning the system state vector may or may not be contained in the observations, at each sampling time, there is a positive probability (called *false alarm probability*) that only noise is observed and, hence, that the observation does not contain the transmitted signal, but it is not generally known whether the observation used for estimation contains the signal or it is only noise. To describe this interrupted observation mechanism (*uncertain observations*), the observation equation, with the usual additive measurement noise, is formulated by multiplying the signal function at each sampling time by a binary random variable taking the values one and zero (Bernoulli random variable); the value one indicates that the measurement at that time contains the signal, whereas the value zero reflects the fact that the signal is missing and, hence, the corresponding observation is only noise. So, the observation equation involves both an additive and a multiplicative noise, the latter modeling

the uncertainty about the signal being present or missing at each observation.

cited.

Linear discrete-time systems with uncertain observations have been widely used in estimation problems related to the above practical situations (which commonly appear, for example, in

> ©2012 Linares-Pérez et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 Linares-Pérez et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
