Uniqueness and stability of nonlinear filtering equations with unbounded random coefficients
Abstract
We study a multidimensional nonlinear filtering model whose coefficients depend on a given observation-adapted predictable process and whose observation drift may grow linearly in both the state and the random input. Due to the unboundedness of the observation drift, a global reference measure is not available. To overcome this hurdle, a localized entropy argument is adapted to prove the stopped likelihood to be a uniformly integrable martingale at each control-energy stopping level. The stopped Zakai equation, and hence, the stopped filtering equation is derived. The global filtering equation is then established by de-localization. The uniqueness of the solution to the stopped Zakai equation is obtained by a duality backward stochastic partial differential equation. This uniqueness then propagates to that of the global filtering equation through the stopped ones. Finally, a stability result is established in W1-distance of measures.
Related
- Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
- Quantitative uniqueness for bi-Laplace equations with C2 lower-order coefficients
- Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
- Well-posedness of Filtering Equations in Weighted Sobolev Spaces with Unbounded System Coefficients
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BibTeX
@misc{xiong2026uniqueness,
title = {{Uniqueness and stability of nonlinear filtering equations with unbounded random coefficients}},
author = {Jie Xiong and Wen Xu and Ying Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28985},
eprint = {2609.28985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28985},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Uniqueness and stability of nonlinear filtering equations with unbounded random coefficients
AU - Jie Xiong
AU - Wen Xu
AU - Ying Yang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28985
UR - https://arxiv.org/abs/2609.28985
AB - We study a multidimensional nonlinear filtering model whose coefficients depend on a given observation-adapted predictable process and whose observation drift may grow linearly in both the state and the random input. Due to the unboundedness of the observation drift, a global reference measure is not available. To overcome this hurdle, a localized entropy argument is adapted to prove the stopped likelihood to be a uniformly integrable martingale at each control-energy stopping level. The stopped Zakai equation, and hence, the stopped filtering equation is derived. The global filtering equation is then established by de-localization. The uniqueness of the solution to the stopped Zakai equation is obtained by a duality backward stochastic partial differential equation. This uniqueness then propagates to that of the global filtering equation through the stopped ones. Finally, a stability result is established in W_1-distance of measures.
ER -