### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 1193-1229 |

Number of pages | 37 |

Journal | Multiscale Modeling and Simulation |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Modeling and Simulation
- Chemistry(all)
- Computer Science Applications
- Ecological Modeling
- Physics and Astronomy(all)

### Cite this

*Multiscale Modeling and Simulation*,

*12*(3), 1193-1229. https://doi.org/10.1137/140952648

**Filtering the maximum likelihood for multiscale problems.** / Papanicolaou, Andrew; Spiliopoulos, Konstantinos.

Research output: Contribution to journal › Article

*Multiscale Modeling and Simulation*, vol. 12, no. 3, pp. 1193-1229. https://doi.org/10.1137/140952648

}

TY - JOUR

T1 - Filtering the maximum likelihood for multiscale problems

AU - Papanicolaou, Andrew

AU - Spiliopoulos, Konstantinos

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Central limit theory

KW - Ergodic filtering

KW - Fast mean reversion

KW - Homogenization

KW - Maximum likelihood estimation

KW - Zakai equation

UR - http://www.scopus.com/inward/record.url?scp=84907938410&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907938410&partnerID=8YFLogxK

U2 - 10.1137/140952648

DO - 10.1137/140952648

M3 - Article

VL - 12

SP - 1193

EP - 1229

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 3

ER -