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Optimal transformations for prediction in continuous-time stochastic processes: finite past and future
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  • Published: 10 February 2005

Optimal transformations for prediction in continuous-time stochastic processes: finite past and future

  • Basilis Gidas1 &
  • Alejandro Murua2 

Probability Theory and Related Fields volume 131, pages 479–492 (2005)Cite this article

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Abstract.

In the classical Wiener-Kolmogorov linear prediction problem, one fixes a linear functional in the ‘‘future’’ of a stochastic process, and seeks its best predictor (in the L2-sense). In this paper we treat a variant of the prediction problem, whereby we seek the ‘‘most predictable’‘ non-trivial functional of the future and its best predictor; we refer to such a pair (if it exists) as an optimal transformation for prediction. In contrast to the Wiener-Kolmogorov problem, an optimal transformation for prediction may not exist, and if it exists, it may not be unique. We prove the existence of optimal transformations for finite ‘‘past’’ and ‘‘future’’ intervals, under appropriate conditions on the spectral density of a weakly stationary, continuous-time stochastic process. For rational spectral densities, we provide an explicit construction of the transformations via differential equations with boundary conditions and an associated eigenvalue problem of a finite matrix.

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Acknowledgments.

We thank one of the referees for bringing Yaglom’s paper to our attention, and the other referee for his constructive remarks. We also thank the referee whose question on equation (2.5) led to the Appendix.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, R.I, 02912, USA

    Basilis Gidas

  2. Department of Statistics, University of Washington, Seattle, WA, 98195-4322, USA

    Alejandro Murua

Authors
  1. Basilis Gidas
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  2. Alejandro Murua
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Corresponding author

Correspondence to Basilis Gidas.

Additional information

This research was partially supported by ARO (MURI grant) DAAH04-96-1-0445, NSF grant DMS-0074276, and CNPq grant 301179/00-0.

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Gidas, B., Murua, A. Optimal transformations for prediction in continuous-time stochastic processes: finite past and future. Probab. Theory Relat. Fields 131, 479–492 (2005). https://doi.org/10.1007/s00440-004-0371-x

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  • Received: 07 December 2001

  • Revised: 30 March 2004

  • Published: 10 February 2005

  • Issue Date: April 2005

  • DOI: https://doi.org/10.1007/s00440-004-0371-x

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Keywords

  • Boundary Condition
  • Differential Equation
  • Stochastic Process
  • Probability Theory
  • Spectral Density
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