# What is the Lagrangian for Nonlinear Filtering?

## TL;DR

This paper extends the classical duality between estimation and control from linear to nonlinear filtering, introducing a dual process via BSDEs to derive the nonlinear filter equation.

## Contribution

It generalizes the Kalman-Bucy duality to nonlinear filters using backward stochastic differential equations and optimal control techniques.

## Key findings

- Derived the nonlinear filter equation using duality and BSDEs.
- Showed classical Kalman-Bucy duality as a special case.
- Provided a new framework for nonlinear filtering via dual processes.

## Abstract

Duality between estimation and optimal control is a problem of rich historical significance. The first duality principle appears in the seminal paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic (LQ) optimal control problem. Duality offers a constructive proof technique to derive the Kalman filter equation from the optimal control solution. This paper generalizes the classical duality result of Kalman-Bucy to the nonlinear filter: The state evolves as a continuous-time Markov process and the observation is a nonlinear function of state corrupted by an additive Gaussian noise. A dual process is introduced as a backward stochastic differential equation (BSDE). The process is used to transform the problem of minimum variance estimation into an optimal control problem. Its solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The classical duality result of Kalman-Bucy is shown to be a special case.