Nonlinear Stochastic Filtering with Volterra Gaussian noises

TL;DR

This paper develops a nonlinear filtering framework for systems driven by Volterra Gaussian rough paths, addressing challenges of irregularity and memory effects, and introduces a rough PDE analogue of the Zakai equation.

Contribution

It establishes well-posedness, robustness, and regularity properties of the nonlinear filter driven by Volterra Gaussian noises, and derives a rough PDE for the filter's density in one dimension.

Findings

Proved well-posedness of rough differential equations with Volterra Gaussian noise.

Established regularity and smoothness of the filter's density.

Derived a rough PDE analogous to the Zakai equation for the unnormalized filter.

Abstract

We consider a nonlinear filtering problem for a signal-observation system driven by a Volterra-type Gaussian rough path, whose sample paths may exhibit greater roughness than those of Brownian motion. The observation process includes a Volterra-type drift, introducing both memory effects and low regularity in the dynamics. We prove well-posedness of the associated rough differential equations and the Kallianpur-Striebel. We then establish robustenss properties of the filter and study the existence, smoothness, and time regularity of its density using partial Malliavin calculus. Finally, we show that, in the one-dimensional case, the density of the unnormalized filter solves a rough partial differential equation, providing a rough-path analogue of the Zakai equation.