More often than not, we cannot directly measure many phenomena that are crucial to us. However, we usually have access to certain partial observations on the phenomena of interest as well as a mathematical model of them. The filtering problem seeks estimation of the phenomena given all the accumulated partial information. In this thesis, we study several topics concerning the numerical approximation of the filtering problem.
First, we study the continuoustime filtering problem. Given highfrequency ob servations in discretetime, we perform double discretization of the nonlinear filter to allow for filter estimation with particle filter. By using the multilevel strategy, given any ε > 0, our algorithm achieve an MSE level of O(ε2) with a cost of O(ε−3), while the particle filter requires a cost of O(ε−4).
Second, we propose a debias scheme for the particle filter under the partially observed diffusion model. The novel scheme is free of innate particle filter bias and discretization bias, through a double randomization method of [14]. Our estimator is perfectly parallel and achieves a similar cost reduction to the multilevel particle filter.
Third, we look at a highdimensional linear Gaussian statespace model in con tinuous time. We propose a novel multilevel estimator which requires a cost of O(ε−2 log(ε)2) compared to ensemble KalmanBucy filters (EnKBFs) which requiresO(ε−3) for an MSE target of O(ε2). Simulation results verify our theory for models of di mension ∼ 106.
Lastly, we consider the model estimation through learning an unknown parameter that characterizes the partially observed diffusions. We propose algorithms to provide unbiased estimates of the Hessian and the inverse Hessian, which allows secondorder
optimization parameter learning for the model.
Date of Award  Mar 13 2022 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Ajay Jasra (Supervisor) 

 particle filtering
 variance reduction
 unbiased estimation
 coupling technique
 kalman filter