Well-Posedness and Negative-Sobolev Stability of a Correlated-Noise Zakai Equation
Abstract
We study a correlated-noise Zakai equation for nonlinear filtering when the signal and observation share a Brownian component. Under bounded coefficients, with bounded-Lipschitz state regularity for the second-order, correlated-noise, and observation coefficients, and under uniform ellipticity of the private signal noise, the equation is well posed in the variational L 2 framework. The main result is a quantitative inhomogeneous H -1 stability estimate based on (I -∆) -1/2 : the second-order drift and the correlated stochastic divergence combine through the parabolicity matrix 2a-γγ T to produce time-integrated L 2 dissipation controlled by the initial H -1 distance. This mechanism also yields stability under L ∞ coefficient perturbations within uniformly bounded-Lipschitz classes, without convergence of coefficient derivatives. A weighted L 2 propagation estimate gives an explicit no-escape criterion, allowing weak L 2 convergence of initial densities to imply the strong negative-topology convergence required by the stability theorem. Under uniform control bounds, the same estimates imply H -1 continuity of separated linear-in-density costs and of the associated value function. We further establish a controlled approximation chain from whole-space localization to stable Galerkin projection, fixed-dimensional Euler-Maruyama time stepping, and quadrature, under the explicit additional temporal and spatial regularity assumptions stated for those numerical steps. Reduced finite-dimensional crowd and traffic models provide theorem-compatible numerical benchmarks and a comparison with a positive weighted-particle Kallianpur-Striebel representation.
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BibTeX
@misc{dogbe2026wellposedness,
title = {{Well-Posedness and Negative-Sobolev Stability of a Correlated-Noise Zakai Equation}},
author = {C. Dogbe},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05747042v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-Posedness and Negative-Sobolev Stability of a Correlated-Noise Zakai Equation
AU - C. Dogbe
PY - 2026
DA - 2026/09/11
PB - HAL
UR - https://hal.science/hal-05747042v1
AB - We study a correlated-noise Zakai equation for nonlinear filtering when the signal and observation share a Brownian component. Under bounded coefficients, with bounded-Lipschitz state regularity for the second-order, correlated-noise, and observation coefficients, and under uniform ellipticity of the private signal noise, the equation is well posed in the variational L 2 framework. The main result is a quantitative inhomogeneous H -1 stability estimate based on (I -∆) -1/2 : the second-order drift and the correlated stochastic divergence combine through the parabolicity matrix 2a-γγ T to produce time-integrated L 2 dissipation controlled by the initial H -1 distance. This mechanism also yields stability under L ∞ coefficient perturbations within uniformly bounded-Lipschitz classes, without convergence of coefficient derivatives. A weighted L 2 propagation estimate gives an explicit no-escape criterion, allowing weak L 2 convergence of initial densities to imply the strong negative-topology convergence required by the stability theorem. Under uniform control bounds, the same estimates imply H -1 continuity of separated linear-in-density costs and of the associated value function. We further establish a controlled approximation chain from whole-space localization to stable Galerkin projection, fixed-dimensional Euler-Maruyama time stepping, and quadrature, under the explicit additional temporal and spatial regularity assumptions stated for those numerical steps. Reduced finite-dimensional crowd and traffic models provide theorem-compatible numerical benchmarks and a comparison with a positive weighted-particle Kallianpur-Striebel representation.
ER -