# Online Sequential Monte Carlo smoother for partially observed stochastic differential equations

## TL;DR

This paper presents an online Monte Carlo smoothing algorithm for partially observed stochastic differential equations, using unbiased estimators to handle unknown transition densities, enabling real-time data processing with linear complexity.

## Contribution

Introduces a novel online smoothing algorithm for SDEs that employs unbiased estimators to manage unknown transition densities, extending previous methods.

## Key findings

- Algorithm is consistent and effective.
- Performance demonstrated on two models.
- Computational complexity grows linearly with samples.

## Abstract

This paper introduces a new algorithm to approximate smoothed additive functionals for partially observed stochastic differential equations. This method relies on a recent procedure which allows to compute such approximations online, i.e. as the observations are received, and with a computational complexity growing linearly with the number of Monte Carlo samples. This online smoother cannot be used directly in the case of partially observed stochastic differential equations since the transition density of the latent data is usually unknown. We prove that a similar algorithm may still be defined for partially observed continuous processes by replacing this unknown quantity by an unbiased estimator obtained for instance using general Poisson estimators. We prove that this estimator is consistent and its performance are illustrated using data from two models.