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  • O. Hijab
Chapter
Part of the Applications of Mathematics book series (SMAP, volume 20)

Abstract

Let (Ω, F, P) be a probability space and let F t , t ≥ 0, be a nondecreasing family of sub-σ-algebras of F. Let η(·) be an (Ω, F t , P) Brownian motion in ℝ m . Let θ: ω → {1,..., N} be F 0-measurable with P(θ = j) = \(P\left( {\theta = j} \right) = \pi _j^0\); here \(\left\{ {\pi _1^0 , \ldots \pi _N^0 } \right\}\) is a fixed but arbitrary distribution on {1,..., N}. For each j = 1,..., N let z j (·) be a progressively measurable process. Throughout this chapter we shall assume that
$$P\left( {\mathop {\max }\limits_{1 \leqslant j \leqslant N} \int_0^T {\left| {z_j \left( t \right)} \right|^2 dt < \infty ,T > 0} } \right) = 1.$$
(1.1)

Keywords

Brownian Motion Kalman Filter Probability Space Stochastic Differential Equation Measurable Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes and References

  1. [4.1]
    D. F. Allinger and S. K. Mitter, “New Results on the Innovations Problem for Nonlinear Filtering,” Stochastics, 4 (1981), 339–348.MathSciNetMATHCrossRefGoogle Scholar
  2. [4.2]
    T. E. Duncan, “On the Calculation of Mutual Information,” SIAM J. Appl. Math., 19 (1970), 215–220.MathSciNetMATHCrossRefGoogle Scholar
  3. [4.3]
    K. P. Dunn, “Measure Transformation, Estimation, Detection, and Stochastic Control,” Ph.D. Dissertation, Washington University, St. Louis, MO, May 1974.Google Scholar
  4. [4.4]
    R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 1984.MATHGoogle Scholar
  5. [4.5]
    M. Fujisaki, G. Kallianpur, and H. Kunita, “Stochastic Differential Equations for the Nonlinear Filtering Problem,” Osaka J. Math., 9 (1972), 19–40.MathSciNetMATHGoogle Scholar
  6. [4.6]
    G. Kallianpur and C. Streibel, “Estimation of Stochastic Processes,” Ann. Math. Statist., 39 (1968), 785–801.MathSciNetMATHCrossRefGoogle Scholar
  7. [4.7]
    J. C. Willems, “Recursive Filtering,” Statist. Neerlandica, 32 (1978), 1–38.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1987

Authors and Affiliations

  • O. Hijab
    • 1
  1. 1.Mathematics DepartmentTemple UniversityPhiladelphiaUSA

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