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.

翻译：我们考虑的是某类连续时间状态空间模型的参数估算问题。 特别是, 我们探索部分观测到的传播, 数据也根据一个扩散过程到达。 基于分数函数的标准身份, 我们考虑两种基于粒子过滤法来估计分数函数。 两种方法都依赖于对 $\ mathcal{O} (N ⁇ 2) 的评分函数的在线估算算法, 以 $N\ in\ mathbb{N} 的粒子数量 。 第一种方法使用简单的 Euler 梯度离异和标准粒子平滑器, 其成本为 $\ mathc{O} (N2+ N\ Delta_ l ⁇ _ l ⁇ -1} 美元 美元/ 单位时间值, $\ Delta_ l=2\\\ l} 美元, 美元= mathb{N ⁇ 0$, 时间分数计算法。 第二种方法是新的, 以新的传播桥构造为基础。 它产生一种新的后向后Feynman- Kac 公式, 在连续时间段计算, 的分数计算函数, 然后用一个时间=xxxxxxxxxxxxx