Filtering the maximum likelihood for multiscale problems

Andrew Papanicolaou, Konstantinos Spiliopoulos

Research output: Contribution to journalArticlepeer-review


Filtering and parameter estimation under partial information for multiscale diffusion problems are studied in this paper. The nonlinear filter converges in the mean-square sense to a filter of reduced dimension. Based on this result, we establish that the conditional (on the observations) log-likelihood process has a correction term given by a type of central limit theorem. We prove that an appropriate normalization of the log-likelihood minus a log-likelihood of reduced dimension converges weakly to a normal distribution. In order to achieve this we assume that the operator of the (hidden) fast process has a discrete spectrum and an orthonormal basis of eigenfunctions. We then propose to estimate the unknown model parameters using the reduced log-likelihood, which is beneficial because reduced dimension means that there is significantly less runtime for this optimization program. We also establish consistency and asymptotic normality of the maximum likelihood estimator. Simulation results illustrate our theoretical findings.

Original languageEnglish (US)
Pages (from-to)1193-1229
Number of pages37
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2014


  • Central limit theory
  • Ergodic filtering
  • Fast mean reversion
  • Homogenization
  • Maximum likelihood estimation
  • Zakai equation

ASJC Scopus subject areas

  • General Chemistry
  • Modeling and Simulation
  • Ecological Modeling
  • General Physics and Astronomy
  • Computer Science Applications


Dive into the research topics of 'Filtering the maximum likelihood for multiscale problems'. Together they form a unique fingerprint.

Cite this