Duality for nonlinear filtering

Kim, Jin Won

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https://hdl.handle.net/2142/116215 Copy

Description

Title

Duality for nonlinear filtering

Author(s)

Kim, Jin Won

Issue Date

2022-07-15

Director of Research (if dissertation) or Advisor (if thesis)

Mehta, Prashant G

Doctoral Committee Chair(s)

Mehta, Prashant G

Committee Member(s)

• Hajek, Bruce
• Raginsky, Maxim
• Dey, Partha S

Department of Study

Mechanical Sci & Engineering

Discipline

Mechanical Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Stochastic filtering
• Backward stochastic differential equations
• Duality

Abstract

This thesis is concerned with the stochastic filtering problem for a hidden Markov model (HMM) with the white noise observation model. For this filtering problem, we make three types of original contributions: (1) dual controllability characterization of stochastic observability, (2) dual minimum variance optimal control formulation of the stochastic filtering problem, and (3) filter stability analysis using the dual optimal control formulation. For the first contribution of this thesis, a backward stochastic differential equation (BSDE) is proposed as the dual control system. The observability (detectability) of the HMM is shown to be equivalent to the controllability (stabilizability) of the dual control system. For the linear-Gaussian model, the dual relationship reduces to classical duality in linear systems theory. The second contribution is to transform the minimum variance estimation problem into an optimal control problem. The constraint is given by the dual control system. The optimal solution is obtained via two approaches: (1) by an application of maximum principle and (2) by the martingale characterization of the optimal value. The optimal solution is used to derive the nonlinear filter. The third contribution is to carry out filter stability analysis by studying the dual optimal control problem. Two approaches are presented through Chapters 7 and 8. In Chapter 7, conditional Poincar\'e inequality (PI) is introduced. Based on conditional PI, various convergence rates are obtained and related to literature. In Chapter 8, the stabilizability of the dual control system is shown to be a necessary and sufficient condition for filter stability on certain finite state space model.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Jin Won Kim

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