The Method of Batch Inference for Multivariate Diffusions

Abstract

Diffusion processes have been used to model a variety of continuous-time phenomena in Finance, Engineering, and the Natural Sciences. However, parametric inference has long been complicated by an intractable likelihood function, the solution of a partial differential equation. For many multivariate models, the most effective inference approach involves a large amount of missing data for which the typical Gibbs sampler can be arbitrarily slow. On the other hand, a recent method of joint parameter and missing data proposals can lead to a radical improvement, but their acceptance rate scales exponentially with the number of observations. We consider here a method of dividing the inference process into separate data batches, each small enough to benefit from joint proposals. A conditional independence argument allows batch-wise missing data to be sequentially integrated out. Although in practice the integration is only approximate, the Batch posterior and the exact parameter posterior can often have similar performance under a Frequency evaluation, for which the true parameter value is fixed. We present an example using Heston’s stochastic volatility model for financial assets, but much of the methodology extends to Hidden Markov and other State-Space models.