Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Xiong, Jie, Xu, Wen, Xu, Zuo Quan, Yang, Ying
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7220985042