Using Random Quasi-Monte-Carlo Within Particle Filters, With Application to Financial Time Series

Abstract

This article presents a new particle filter algorithm which uses random quasi-Monte-Carlo to propagate particles. The filter can be used generally, but here it is shown that for one-dimensional state-space models, if the number of particles is N, then the rate of convergence of this algorithm is N⁻¹. This compares favorably with the N^−1/2 convergence rate of standard particle filters. The computational complexity of the new filter is quadratic in the number of particles, as opposed to the linear computational complexity of standard methods. I demonstrate the new filter on two important financial time series models, an ARCH model and a stochastic volatility model. Simulation studies show that for fixed CPU time, the new filter can be orders of magnitude more accurate than existing particle filters. The new filter is particularly efficient at estimating smooth functions of the states, where empirical rates of convergence are N^−3/2; and for performing smoothing, where both the new and existing filters have the same computational complexity.

Key Words:

• ARCH
• Filtering
• Rate of convergence
• Sequential Monte Carlo
• Smoothing
• Stochastic volatility

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