INS seminars 58: The adaptive patched particle filter and its implementation (Wonjung Lee, May.02, 2013)

INS seminars 58

Abstract:

The Kusuoka-Lyons-Victoir (KLV) approach is a higher order particle method for approximating the weak solution of a stochastic differential equation. The algorithm can be performed by integrating along a number of carefully selected bounded variation paths and the iterated application of the KLV method has a tendency for the number of particles to increase. Together with local dynamic recombination that simplifies the support of discrete measure without harming the accuracy of the approximation, the KLV method becomes eligible to solve the filtering problem for which one has to maintain an accurate description of the ever-evolving conditioned measure. Besides the alternate application of the KLV method and recombination for the entire family of particles, we make use of the smooth nature of likelihood to lead some of the particles immediately to the next observation time and to build an algorithm that is a form of automatic high order adaptive importance sampling. We perform numerical simulations to evaluate the efficiency and accuracy of the proposed approaches in the example of the linear stochastic differential equation driven by three independent Brownian motions. Our numerical simulations show that even when the sequential Monte-Carlo method poorly performs, the KLV method and recombination can together be used to approximate higher order moments of the filtering solution in a moderate dimension with high accuracy and efficiency.