Increasing domain infill asymptotics for stochastic differential equations driven by fractional Brownian motion

Although statistical inference in stochastic differential equations (<i>SDE</i>s) driven by Wiener process has received significant attention in the literature, inference in those driven by fractional Brownian motion seems to have seen much less development in comparison, despite their importance in modelling long-range dependence. In this article, we consider both classical and Bayesian inference in such fractional Brownian motion-based <i>SDE</i>s, observed on the time domain [0,T]. In particular, we consider asymptotic inference for two parameters in this regard; a multiplicative parameter <i>β</i> associated with the drift function, and the so called ‘Hurst parameter’ <i>H</i> of the fractional Brownian motion, when T→∞. For unknown <i>H</i>, 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 <i>SDE</i> into <i>n</i> discrete time points in [0,T], and letting T→∞, n→∞, such that either n/T2 or <i>n</i>/<i>T</i> tends to infinity. 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 <i>SDE</i>s driven by fractional Brownian motion. Applications to a real, close price data, along with comparison with standard <i>SDE</i> driven by Wiener process, are also considered. As expected, it turned out that our Bayesian fractional <i>SDE</i> outperformed the other model and methods, in both simulated and real data applications.