Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models

TL;DR

This paper introduces two particle filter-based methods for parameter estimation in continuous-time state space models, including a novel diffusion bridge approach, with a focus on computational efficiency and practical implementation.

Contribution

It presents a new diffusion bridge-based particle method for score estimation and explores multilevel techniques to reduce computational costs.

Findings

The new diffusion bridge approach provides a continuous-time Feynman-Kac formula for the score.

Multilevel methods improve computational efficiency for parameter estimation.

Numerical examples demonstrate the effectiveness of the proposed methods.

Abstract

We consider the problem of parameter estimation for a class of continuous-time state space models. In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time, where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward type Feynman-Kac formula in continuous-time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.