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

We consider the problem of parameter estimation for a class continuous-time state space models (SSMs). In particular, we explore case partially observed diffusion, with data also arriving according to diffusion process. Based upon standard identity score function, two particle filter based methodologies estimate function. Both methods rely on an online algorithm as described, e.g., in [P. Del Moral, A. Doucet, and S. Singh, M$2$AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], $\mathcal{O}(N^2)$ cost, $N\in\mathbb{N}$ number particles. The first approach employs simple Euler discretization smoothers is cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time, where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, time-discretization step. second new novel bridge construction. It yields backward-type Feynman--Kac formula continuous time function presented along method its approximation. Considering time-discretization, $\mathcal{O}(N^2\Delta_l^{-1})$ time. To improve computational costs, then multilevel illustrate our via stochastic gradient approaches several numerical examples.