Application of Fractional Brownian Motion (fBm) and Hurst Exponent Analysis in Financial Modeling: A Biophysics-Based FFT–MCMC Method

Fractional Brownian motion (fBm) provides a powerful stochastic framework for modeling long-range temporal dependence that cannot be represented by classical Brownian motion. This study presents a numerical and theoretical investigation of constrained fractional Brownian motion with applications to stochastic financial systems. An efficient simulation framework combining Fast Fourier Transform (FFT)-based circulant embedding and Markov Chain Monte Carlo (MCMC) sampling is developed to generate long correlated trajectories under absorbing boundary conditions. The proposed algorithm enables simulations with trajectory lengths up to L = 107 while reducing the computational complexity from O (L3) for direct covariance decomposition to approximately O(L log L). Numerical results accurately reproduce the theoretical autocorrelation function of fBm and confirm the expected persistence behavior governed by the Hurst exponent. Super-diffusive regimes (H > 0.5) exhibit persistent long-range correlations and enhanced survival probabilities, whereas sub-diffusive regimes (H < 0.5) display anti-persistent dynamics and increased boundary absorption. The fractional stochastic volatility formulation captures important characteristics associated with long-memory financial systems, including persistent volatility dynamics and implied-volatility structures. The proposed biophysical-based FFT–MCMC methodology provides an accurate, scalable, and computationally efficient framework for studying constrained fractional stochastic processes and offers a foundation for future investigations of fractional volatility models and related financial applications. A conceptual Adaptive Hurst Momentum framework is briefly discussed as a possible direction for future research.