# Joint Maximum a Posteriori State Path and Parameter Estimation in Stochastic Differential Equations

## TL;DR

This paper introduces the joint maximum a posteriori estimator for state and parameter estimation in stochastic differential equations, demonstrating improved robustness and lower errors compared to traditional methods in simulations.

## Contribution

The paper presents a novel joint MAP estimator for continuous-time SDE systems that handles nonlinearities and discrete measurements, extending existing estimation techniques.

## Key findings

- JME outperforms PEM in state estimation accuracy with outliers.
- MEE is biased for certain damping parameters.
- JME provides more robust estimates in simulated experiments.

## Abstract

In this article, we introduce the joint maximum a posteriori state path and parameter estimator (JME) for continuous-time systems described by stochastic differential equations (SDEs). This estimator can be applied to nonlinear systems with discrete-time (sampled) measurements with a wide range of measurement distributions. We also show that the minimum-energy state path and parameter estimator (MEE) obtains the joint maximum a posteriori noise path, initial conditions, and parameters. These estimators are demonstrated in simulated experiments, in which they are compared to the prediction error method (PEM) using the unscented Kalman filter and smoother. The experiments show that the MEE is biased for the damping parameters of the drift function. Furthermore, for robust estimation in the presence of outliers, the JME attains lower state estimation errors than the PEM.