Remarks on the finite energy condition in additive white noise filtering

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For the nonlinear filtering problem y (t) = ∫0t h (x (s)) d s + w (t), the finite energy condition, Ε ∫0T h2 (x (s)) d s < ∞, and state-observation noise independence imply that the martingale L (t) = exp {∫0t h (x (s)) d y (s) - frac(1, 2) ∫0t h2 (x (s)) d s} is an H1-martingale with respect to the transformed measure. This fact is used to obtain a direct proof of Zakai's equation and to prove that the total mass of the unnormalized conditional density is also an H1-martingale.

Original languageEnglish (US)
Pages (from-to)197-203
Number of pages7
JournalSystems and Control Letters
Issue number3
StatePublished - Dec 1984

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science(all)
  • Mechanical Engineering
  • Electrical and Electronic Engineering


  • H-martingales
  • Nonlinear filtering
  • Zakai's equation


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