Backward stochastic partial differential equations with unbounded random coefficients
Abstract
We study a linear backward stochastic partial differential equation with unbounded observation-adapted random coefficients, motivated by the duality approach to nonlinear filtering. The coefficients depend predictably on the observation history through a control with almost surely finite time-integrated squared norm. The drift, diffusion, and coefficient multiplying the martingale integrand may grow linearly in space. The diffusion matrix may grow quadratically in space and is allowed to be degenerate. Under bounded positive-order spatial derivatives and bounded smooth observation-measurable terminal data, we prove existence and uniqueness in a polynomially weighted Sobolev class up to control-energy stopping times. The proof combines conditional parameter estimates, entropy bounds for likelihoods, stability under the reference probability, and stochastic-flow reconstruction.
Related
- Multidimensional Infinite-Horizon Markovian Backward Stochastic Differential Equations with Continuous Coefficients and Nonzero-Sum Differential Games
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- Backward doubly stochastic differential equations with or without reflection under weak conditions
- Conditional Reflected Backward Stochastic Differential Equations
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BibTeX
@misc{xiong2026backward,
title = {{Backward stochastic partial differential equations with unbounded random coefficients}},
author = {Jie Xiong and Wen Xu and Zuo Quan Xu and Ying Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09278},
eprint = {2610.09278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09278},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Backward stochastic partial differential equations with unbounded random coefficients
AU - Jie Xiong
AU - Wen Xu
AU - Zuo Quan Xu
AU - Ying Yang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.09278
UR - https://arxiv.org/abs/2610.09278
AB - We study a linear backward stochastic partial differential equation with unbounded observation-adapted random coefficients, motivated by the duality approach to nonlinear filtering. The coefficients depend predictably on the observation history through a control with almost surely finite time-integrated squared norm. The drift, diffusion, and coefficient multiplying the martingale integrand may grow linearly in space. The diffusion matrix may grow quadratically in space and is allowed to be degenerate. Under bounded positive-order spatial derivatives and bounded smooth observation-measurable terminal data, we prove existence and uniqueness in a polynomially weighted Sobolev class up to control-energy stopping times. The proof combines conditional parameter estimates, entropy bounds for likelihoods, stability under the reference probability, and stochastic-flow reconstruction.
ER -