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Abstract
In this paper we consider continuous-time diffusion models driven by frac- tional
Brownian Motion (fBM), with observations obtained at discrete-time instances. As a
prototypical scenario we will give emphasis on a a stochas- tic volatility (SV) model
allowing for memory in the volatility increments through an fBM specification. Due
to the non-Markovianity of the model and the high-dimensionality of the latent volatility
path, estimating posterior expectations is a computationally challenging task. We
present novel sim- ulation and re-parameterisation framework based on the Davies and
Harte method and use it to construct a Markov chain Monte-Carlo (MCMC) algo- rithm
that allows for computationally efficient parametric Bayesian inference upon application
on such models. The algorithm is based on an advanced version of the so-called Hybrid
Monte-Carlo (HMC) that allows for increased efficiency when applied on high-dimensional
latent variables relevant to the models of interest in this paper. The inferential
methodology is examined and illustrated in the SV models, on simulated data as well
as real data from the S&P500/VIX time series. Contrary to a long range dependence
attribute of the SV process (Hurst parameter H > 1/2) many times assumed in the literature,
the posterior distribution favours H < 1/2 that points towards medium range dependence.