  Although statistical inference in stochastic differential equations (SDEs)
driven by Wiener process has received significant attention in the literature,
inference in those driven by fractional Brownian motion seem to have seen much
less development in comparison, despite their importance in modeling long range
dependence. In this article, we consider both classical and Bayesian inference
in such fractional Brownian motion based SDEs. In particular, we consider
asymptotic inference for two parameters in this regard; a multiplicative
parameter associated with the drift function, and the so-called "Hurst
parameter" of the fractional Brownian motion, when the time domain tends to
infinity. For unknown Hurst parameter, the likelihood does not lend itself
amenable to the popular Girsanov form, rendering usual asymptotic development
difficult. As such, we develop increasing domain infill asymptotic theory, by
discretizing the SDE. In this setup, we establish consistency and asymptotic
normality of the maximum likelihood estimators, as well as consistency and
asymptotic normality of the Bayesian posterior distributions. However,
classical or Bayesian asymptotic normality with respect to the Hurst parameter
could not be established. We supplement our theoretical investigations with
simulation studies in a non-asymptotic setup, prescribing suitable
methodologies for classical and Bayesian analyses of SDEs driven by fractional
Brownian motion. Applications to a real, close price data, along with
comparison with standard SDE driven by Wiener process, is also considered. As
expected, it turned out that our Bayesian fractional SDE triumphed over the
other model and methods, in both simulated and real data applications.
