Likelihood-based inference for correlated diffusions

TL;DR

This paper develops a likelihood-based inference method for correlated diffusion processes using MCMC, addressing positive definiteness constraints and likelihood inaccessibility, with applications to financial data.

Contribution

It introduces a novel MCMC scheme with Cholesky factorisation for correlated diffusions and extends data augmentation to multivariate stochastic volatility models.

Findings

Effective inference demonstrated on simulated data

Application to real financial exchange rate data

Method handles high-dimensional diffusion models

Abstract

We address the problem of likelihood based inference for correlated diffusion processes using Markov chain Monte Carlo (MCMC) techniques. Such a task presents two interesting problems. First, the construction of the MCMC scheme should ensure that the correlation coefficients are updated subject to the positive definite constraints of the diffusion matrix. Second, a diffusion may only be observed at a finite set of points and the marginal likelihood for the parameters based on these observations is generally not available. We overcome the first issue by using the Cholesky factorisation on the diffusion matrix. To deal with the likelihood unavailability, we generalise the data augmentation framework of Roberts and Stramer (2001 Biometrika 88(3):603-621) to d-dimensional correlated diffusions including multivariate stochastic volatility models. Our methodology is illustrated through simulation based experiments and with daily EUR /USD, GBP/USD rates together with their implied volatilities.