# Data assimilation in price formation

## TL;DR

This paper develops a method to estimate buyer and vendor densities in a nonlinear price formation model using measurements, formulating it as an optimal control problem with stability analysis and numerical validation.

## Contribution

It introduces a novel approach to estimate densities in price formation models through optimal control, including stability estimates and numerical experiments.

## Key findings

- Stability estimates for controls and densities are established.
- Numerical experiments support the theoretical results.
- The method effectively estimates densities from measurement data.

## Abstract

We consider the problem of estimating the density of buyers and vendors in a nonlinear parabolic price formation model using measurements of the price and the transaction rate. Our approach is based on a work by Puel et al., see \cite{Puel2002}, and results in a optimal control problem. We analyse this problems and provide stability estimates for the controls as well as the unknown density in the presence of measurement errors. Our analytic findings are supported with numerical experiments.