# State and Parameter Estimation from Observed Signal Increments

## TL;DR

This paper develops ensemble Kalman-Bucy filter algorithms for simultaneous state and parameter estimation in continuous-time stochastic systems with correlated errors, demonstrated on complex multi-scale models.

## Contribution

It introduces new ensemble Kalman-Bucy algorithms tailored for joint state and parameter estimation in correlated error settings.

## Key findings

- Effective estimation in multi-scale stochastic models
- Algorithms handle correlated model and measurement errors
- Successful application to complex systems

## Abstract

The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.