\bfS \bfC \bfO \bfR \bfE -\bfB \bfA \bfS \bfE \bfD \bfP \bfA \bfR \bfA \bfM \bfE \bfT \bfE \bfR \bfE \bfS \bfT \bfI \bfM \bfA \bfT \bfI \bfO \bfN \bfF \bfO \bfR \bfA \bfC \bfL \bfA \bfS \bfS \bfO \bfF \bfC \bfO \bfN \bfT \bfI \bfN \bfU \bfO \bfU \bfS -\bfT \bfI \bfM \bfE \bfS \bfT \bfA \bfT \bfE \bfS \bfP \bfA \bfC \bfE \bfM \bfO \bfD \bfE \bfL \bfS \ast ALEXANDROS BESKOS \dagger , DAN CRISAN \ddagger , AJAY JASRA \S , NIKOLAS KANTAS \ddagger , \mathrm{\mathrm{\mathrm{ HAMZA RUZAYQAT \S \bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . 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, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], of \scrO (N 2 ) cost, with N \in \BbbN the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost \scrO (N 2 + N \Delta - 1 l ) per unit time, where \Delta l = 2 - l , l \in \BbbN 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 \scrO (N 2 \Delta - 1 l ) 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.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . score function, particle filter, diffusion bridges, parameter estimation \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs .

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman-Bucy filter based estimates based upon several nonlinear Kalman-Bucy diffusions. Based upon new conditional bias results for the mean of the afore-mentioned methods, we analyze the empirical log-scale normalization constants in terms of their L n −errors and conditional bias. Depending on the type of nonlinear Kalman-Bucy diffusion, we show that these are of order (t 1/2 /N 1/2 )+t/N or 1/N 1/2 (L n −errors) and of order [t+t 1/2 ]/N or 1/N (conditional bias), where t is the time horizon and N is the ensemble size. Finally, we use these results for online static parameter estimation for above filtering models and implement the methodology for both linear and nonlinear models.

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AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.