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Adaptive Kalman Filter for Systems with Unknown Initial Values

Yury A Kutoyants

TL;DR

This paper develops a recurrent, online framework for adaptive Kalman filtering of partially observed continuous-time linear systems with unknown initial values and/or parameters. It introduces a four-step scheme: a learning-interval-based preliminary estimator, a recurrent One-step MLE-process, and substitution into Kalman–Bucy filtration to obtain adaptive filters, with explicit recurrence forms and asymptotic analyses. In deterministic-initial-value models, the approach achieves asymptotic efficiency for the state estimator, while in random-initial-value models it yields a locally asymptotically mixed normal (LAMN) structure and a provably consistent One-step MLE, albeit with discussions on efficiency limits. The work emphasizes online implementability and provides rigorous asymptotic characterizations of estimation error and filter performance, offering a pathway to practical, computationally light adaptive filtering in continuous-time hidden-state models.

Abstract

The models of partially observed linear stochastic differential equations with unknown initial values of the non-observed component are considered in two situations. In the first problem, the initial value is deterministic, and in the second problem, it is assumed to be a Gaussian random variable. The main problem is the computation of adaptive Kalman filters and the discussion of their asymptotic optimality. The realization of this program for both models is done in several steps. First, a preliminary estimator of the unknown parameter is constructed by observations on some learning interval. Then, this estimator is used for the calculation of recurrent one-step MLE estimators, which are subsequently substituted in the equations of Kalman filtration.

Adaptive Kalman Filter for Systems with Unknown Initial Values

TL;DR

This paper develops a recurrent, online framework for adaptive Kalman filtering of partially observed continuous-time linear systems with unknown initial values and/or parameters. It introduces a four-step scheme: a learning-interval-based preliminary estimator, a recurrent One-step MLE-process, and substitution into Kalman–Bucy filtration to obtain adaptive filters, with explicit recurrence forms and asymptotic analyses. In deterministic-initial-value models, the approach achieves asymptotic efficiency for the state estimator, while in random-initial-value models it yields a locally asymptotically mixed normal (LAMN) structure and a provably consistent One-step MLE, albeit with discussions on efficiency limits. The work emphasizes online implementability and provides rigorous asymptotic characterizations of estimation error and filter performance, offering a pathway to practical, computationally light adaptive filtering in continuous-time hidden-state models.

Abstract

The models of partially observed linear stochastic differential equations with unknown initial values of the non-observed component are considered in two situations. In the first problem, the initial value is deterministic, and in the second problem, it is assumed to be a Gaussian random variable. The main problem is the computation of adaptive Kalman filters and the discussion of their asymptotic optimality. The realization of this program for both models is done in several steps. First, a preliminary estimator of the unknown parameter is constructed by observations on some learning interval. Then, this estimator is used for the calculation of recurrent one-step MLE estimators, which are subsequently substituted in the equations of Kalman filtration.

Paper Structure

This paper contains 10 sections, 4 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Model of observations
  3. Deterministic initial value
  4. Unknown initial value
  5. Unknown initial value and parameter
  6. Adaptive filter
  7. Random initial value
  8. Parameter estimators
  9. Adaptive filter
  10. Discussion

Key Result

Proposition 1

Suppose that the conditions ${\cal A}_*$ and ${\cal B}_1-{\cal B}_3$ are satisfied. Then the One-step MLE-process $\vartheta _{t,\varepsilon }^\star,\tau <t\leq T$ is consistent, asymptotically normal and the process $\eta _{t,\varepsilon }\left(\vartheta _0\right),\tau _*\leq t\leq T$ for any $\tau _*\in (\tau ,T]$ converges in distribution in $\left({\cal C}\left(0,T\right),{\mathfrak B}\right)

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • Remark 1
  • Remark 2
  • Proposition 3
  • proof