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Rough stochastic filtering

Fabio Bugini, Peter K. Friz, Khoa Lê, Huilin Zhang

TL;DR

The paper develops a comprehensive rough-path framework for nonlinear stochastic filtering, proving well-posedness (existence, uniqueness, stability) of rough Zakai and rough Kushner--Stratonovich equations under dimension-independent regularity. It extends filtering to rough dynamics, including a rough Kalman--Bucy theory with a rough Riccati equation, and establishes a rigorous bridge to classical stochastic filtering via randomization of the rough path, showing that rough objects converge to their stochastic analogues under the Itô lift. This pathwise formulation yields robust representations and potential applicability to high-dimensional or infinite-dimensional filtering problems, while preserving Gaussian structure in linear cases. The results unify rough analysis with classical filtering theory and provide a robust, conceptually unified approach to foundational problems in stochastic filtering.

Abstract

This article is concerned with the well-posedness of the "filtering equations", due to Zakai and Kushner-Stratonovich, arising in nonlinear stochastic filtering. In general situations, notably in correlated diffusion models and when signal coefficients depend on the observation process, the well-posedness is a difficult problem, mainly due to conflicting martingale structures of the involved forward and backward equations. Crisan-Pardoux (2024) address this classical problem with BSPDE techniques, Du et al. (2013), a Sobolev-based approach that however requires increasingly strong regularity assumptions in high dimensions. In this work, we take a new mixed rough stochastic perspective which allows us to derive well-posed rough counterparts of the filtering equations. Importantly, the rough filtering equations are seen, upon randomization, to coincide with the classical filtering equations. Our framework yields well-posedness (existence, uniqueness, stability) under dimension-independent regularity assumptions, providing a robust and conceptually unified solution to a longstanding problem in stochastic filtering theory. To illustrate the flexibility of the method, we also treat rough versions of the classical Kalman-Bucy filter, with characteristics described by a new class of RDEs of rough Riccati type.

Rough stochastic filtering

TL;DR

The paper develops a comprehensive rough-path framework for nonlinear stochastic filtering, proving well-posedness (existence, uniqueness, stability) of rough Zakai and rough Kushner--Stratonovich equations under dimension-independent regularity. It extends filtering to rough dynamics, including a rough Kalman--Bucy theory with a rough Riccati equation, and establishes a rigorous bridge to classical stochastic filtering via randomization of the rough path, showing that rough objects converge to their stochastic analogues under the Itô lift. This pathwise formulation yields robust representations and potential applicability to high-dimensional or infinite-dimensional filtering problems, while preserving Gaussian structure in linear cases. The results unify rough analysis with classical filtering theory and provide a robust, conceptually unified approach to foundational problems in stochastic filtering.

Abstract

This article is concerned with the well-posedness of the "filtering equations", due to Zakai and Kushner-Stratonovich, arising in nonlinear stochastic filtering. In general situations, notably in correlated diffusion models and when signal coefficients depend on the observation process, the well-posedness is a difficult problem, mainly due to conflicting martingale structures of the involved forward and backward equations. Crisan-Pardoux (2024) address this classical problem with BSPDE techniques, Du et al. (2013), a Sobolev-based approach that however requires increasingly strong regularity assumptions in high dimensions. In this work, we take a new mixed rough stochastic perspective which allows us to derive well-posed rough counterparts of the filtering equations. Importantly, the rough filtering equations are seen, upon randomization, to coincide with the classical filtering equations. Our framework yields well-posedness (existence, uniqueness, stability) under dimension-independent regularity assumptions, providing a robust and conceptually unified solution to a longstanding problem in stochastic filtering theory. To illustrate the flexibility of the method, we also treat rough versions of the classical Kalman-Bucy filter, with characteristics described by a new class of RDEs of rough Riccati type.

Paper Structure

This paper contains 25 sections, 31 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Rough stochastic filtering
  4. Rough stochastic dynamics of signal and observation
  5. Rough Kallianpur--Striebel
  6. Rough Zakai equation in spaces of measures: existence
  7. Uniqueness
  8. Rough Kushner--Stratonovich equation
  9. Extension to degenerate observation noise
  10. Rough Kalman--Bucy theory
  11. Stochastic theory
  12. Rough Riccati equation
  13. Rough Kalman--Bucy filtering
  14. Connecting rough and stochastic filtering
  15. Stochastic vs rough Kallianpur--Striebel formula
  16. ...and 10 more sections

Key Result

Theorem 3.1

Let $X^\mathbf{Y}_0 \in L_0(\Omega,\mathcal{F}_0;\mathbb{R}^{d_X})$. Under assum:E, there exists a unique $L_{2,\infty}$-integrable solution $X^\mathbf{Y}$ to equation eq:roughSDE_filtering in the sense of def:integrablesolutionsRSDEs and also a unique $L_2$-integrable solution $Z^\mathbf{Y}$ to the

Theorems & Definitions (71)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Definition 3.6: Rough Zakai
  • Proposition 3.7
  • proof
  • ...and 61 more