# On the anticipative nonlinear filtering problem and its stability

## TL;DR

This paper introduces a novel anticipative nonlinear filtering framework where observation noise correlates with past signals, applicable in finance and engineering, with derived equations and numerical validation.

## Contribution

It develops new filtering equations for anticipative models, including nonlinear, linear, and finite Volterra-type cases, expanding the scope of nonlinear filtering theory.

## Key findings

- Derived new filter equations for anticipative models
- Validated the approach with numerical experiments
- Extended filtering theory to Volterra-type observations

## Abstract

In this paper, we consider an anticipative nonlinear filtering problem, in which the observation noise is correlated with the past of the signal. This new signal-observation model has its applications in both finance models with insider trading and in engineering. We derive a new equation for the filter in this context, analyzing both the nonlinear and the linear cases. We also handle the case of a finite filter with Volterra type observation. The performance of our algorithm is presented through numerical experiments.