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

Alexandros Beskos, Dan Crisan, Ajay Jasra, Nikolas Kantas, Hamza Mahmoud Ruzayqat

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

## Abstract

We consider the problem of parameter estimation for a class of continuous-time state space models (SSMs). 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, as described, e.g., in [P. Del Moral, A. Doucet, and S. S. Singh, M$2$AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], 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.
Original language English (US) A2555-A2580 1 SIAM Journal on Scientific Computing 43 4 https://doi.org/10.1137/20m1362942 Published - Jul 15 2021

### Bibliographical note

KAUST Repository Item: Exported on 2021-11-21
Acknowledgements: The work of the third and fifth authors was supported by KAUST baseline funding. The work of the first author was supported by a Leverhulme Trust Prize. The work of the second author was partially supported by EU Synergy project STUOD - DLV-856408. The work of the fourth author was supported by a JP Morgan A.I. Faculty award.

## ASJC Scopus subject areas

• Computational Mathematics
• Applied Mathematics

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