**7. References**


[8] Huang, R. & Záruba, G. [2007]. Incorporating data from multiple sensors for localizing nodes in mobile ad hoc networks, *IEEE Transactions on Mobile Computing* 6(No. 9): 1090–1104.

20 Will-be-set-by-IN-TECH

problems of arbitrary degree for such linear systems with uncertain observations correlated in instants that differ *m* units of time. On the other hand, in practical engineering, some recent progress on the filtering and control problems for nonlinear stochastic systems with uncertain observations is being achieved. Nonlinearity and stochasticity are two important sources that are receiving special attention in research and, therefore, filtering and smoothing problems for nonlinear systems with uncertain observations would be relevant topics on which further

This research is supported by Ministerio de Educación y Ciencia (Programa FPU and grant

*Dpto. de Estadística e I.O. Universidad de Granada. Avda. Fuentenueva. 18071. Granada, Spain*

*Dpto. de Estadística e I.O. Universidad de Granada. Avda. Fuentenueva. 18071. Granada, Spain*

[1] Caballero-Águila, R., Hermoso-Carazo, A. & Linares-Pérez, J. [2011]. Linear and quadratic estimation using uncertain observations from multiple sensors with correlated

[2] García-Garrido, I., Linares-Pérez, J., Caballero-Águila, R. & Hermoso-Carazo, A. [2012]. A solution to the filtering problem for stochastic systems with multi-sensor uncertain

[3] Hermoso-Carazo, A. & Linares-Pérez, J. [1994]. Linear estimation for discrete-time systems in the presence of time-correlated disturbances and uncertain observations, *IEEE*

[4] Hermoso-Carazo, A. & Linares-Pérez, J. [1995]. Linear smoothing for discrete-time systems in the presence of correlated disturbances and uncertain observations, *IEEE*

[5] Hermoso-Carazo, A., Linares-Pérez, J., Jiménez-López, J., Caballero-Águila, R. & Nakamori, S. [2008]. Recursive fixed-point smoothing algorithm from covariances based on uncertain observations with correlation in the uncertainty, *Applied Mathematics and*

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*Dpto. de Estadística e I.O. Universidad de Jaén. Paraje Las Lagunillas. 23071. Jaén, Spain*

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No. MTM2011-24718) and Junta de Andalucía (grant No. P07-FQM-02701).

investigation would be interesting.

**Acknowledgements** 

**Author details**

R. Caballero-Águila

I. García-Garrido

**7. References**

J. Linares-Pérez

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**On Guaranteed Parameter Estimation of**

**Stochastic Delay Differential Equations**

Assume (Ω, F,(F(*t*), *t* ≥ 0), *P*) is a given filtered probability space and *W* = (*W*(*t*), *t* ≥ 0), *V* = (*V*(*t*), *t* ≥ 0) are real-valued standard Wiener processes on (Ω, F,(F(*t*), *t* ≥ 0), *P*), adapted to (F(*t*)) and mutually independent. Further assume that *X*<sup>0</sup> = (*X*0(*t*), *t* ∈ [−1, 0]) and *Y*<sup>0</sup> are a real-valued cadlag process and a real-valued random variable on

<sup>0</sup> (*s*)*ds* <sup>&</sup>lt; <sup>∞</sup> and *EY*<sup>2</sup>

Assume *Y*<sup>0</sup> and *X*0(*s*) are F0−measurable, *s* ∈ [−1, 0] and the quantities *W*, *V*, *X*<sup>0</sup> and *Y*<sup>0</sup> are

Consider a two–dimensional random process (*X*,*Y*)=(*X*(*t*),*Y*(*t*), *t* ≥ 0) described by the

with the initial conditions *X*(*t*) = *X*0(*t*), *t* ∈ [−1, 0], and *Y*(0) = *Y*0. The process *X* is supposed to be hidden, i.e., unobservable, and the process *Y* is observed. Such models are used in applied problems connected with control, filtering and prediction of stochastic processes (see,

The parameter *ϑ* = (*a*, *b*)� ∈ Θ is assumed to be unknown and shall be estimated based on continuous observation of *<sup>Y</sup>*, <sup>Θ</sup> is a subset of <sup>R</sup><sup>2</sup> ((*a*, *<sup>b</sup>*)� denotes the transposed (*a*, *<sup>b</sup>*)). Equations (1) and (2) together with the initial values *X*0(·) and *Y*<sup>0</sup> respectively have uniquely

<sup>0</sup> < ∞.

**Chapter 2**

*dX*(*t*) = *aX*(*t*)*dt* + *bX*(*t* − 1)*dt* + *dW*(*t*), (1)

©2012 Küchler and Vasiliev, 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

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.

©2012 Küchler and Vasiliev, licensee InTech. This is a paper distributed under the terms of the Creative Commons

*dY*(*t*) = *X*(*t*)*dt* + *dV*(*t*), *t* ≥ 0 (2)

**by Noisy Observations**

Uwe Küchler and Vyacheslav A. Vasiliev

(Ω, F,(F(*t*), *t* ≥ 0), *P*) respectively with

system of stochastic differential equations

for example, [1, 4, 17–20] among others).

solutions *X*(·) and *Y*(·), for details see [19].

properly cited.

*E* 0 −1 *X*2

http://dx.doi.org/10.5772/45952

**1. Introduction**

mutually independent.

Additional information is available at the end of the chapter

22 Stochastic Modeling and Control **Chapter 0 Chapter 2**
