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


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
$$P\left( {\mathop {\max }\limits_{1 \leqslant j \leqslant N} \int_0^T {\left| {z_j \left( t \right)} \right|^2 dt < \infty ,T > 0} } \right) = 1.$$


Brownian Motion Kalman Filter Probability Space Stochastic Differential Equation Measurable Process 


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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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