The adaptive patched cubature filter and its implementation

There are numerous contexts where one wishes to describe the state of a randomly evolving system. Effective solutions combine models that quantify the underlying uncertainty with available observational data to form scientifically reasonable estimates for the uncertainty in the system state. Stochastic differential equations are often used to mathematically model the underlying system. The Kusuoka-Lyons-Victoir (KLV) approach is a higher-order particle method for approximating the weak solution of a stochastic differential equation that uses a weighted set of scenarios to approximate the evolving probability distribution to a high-order of accuracy. The algorithm can be performed by integrating along a number of carefully selected bounded variation paths. The iterated application of the KLV method has a tendency for the number of particles to increase. This can be addressed and, together with local dynamic recombination, which simplifies the support of discrete measure without harming the accuracy of the approximation, the KLV method becomes eligible to solve the filtering problem in contexts where one desires to maintain an accurate description of the ever-evolving conditioned measure. In addition to the alternate application of the KLV method and recombination, we make use of the smooth nature of the likelihood function and high order accuracy of the approximations to lead some of the particles immediately to the next observation time and to build into the algorithm 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-dimensional Brownian motion. Our numerical simulations show that, even when the sequential Monte-Carlo methods poorly perform, 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.