Yuming Kuang from Stanford University.
Paper: Adaptive Particle Filters in Hidden Markov Models: A New Approach and its Applications
Particle filters, also known as sequential Monte Carlo methods have been widely used to solve the latent state filtering problem in nonlinear hidden Markov model (HMM). In this thesis, we propose a new methodological advancement, adaptive particle filter, for joint parameter estimation and latent state filtering of HMM. Adaptive particle filter is a hybrid algorithm that merges particle filters and a new MCMC scheme, sequential substitution, to provide an efficient estimate for function of the posterior distribution of parameter and latent state, and further give estimator of its standard error. We establish the asymptotic normality of the estimate for the function and the consistency of the standard error estimator. In the case of a long sequence of HMM, we propose the Markov chain restart strategy which enables the particle filters method for newly proposed parameter atoms at time t+δ t to start at time t by utilize the approximation of posterior distribution at time t. Markov chain restart greatly reduces the computation cost of adaptive particle filters and makes it feasible to perform more sequential substitution iterations for long sequence of HMM. We demonstrate the effectiveness of adaptive particle filters and Markov chain restart strategy with simulation results on examples of HMM. As an application in finance and econometrics, we apply our approach on parameter estimation and latent volatility filtering for the jump-diffusion models using the asset returns and option prices.