Stochastic Differential Equation Models for Bitcoin: Rough Volatility, Jumps, and Parsimonious Model Reduction

We aim to identify a parsimonious stochastic differential equation for Bitcoin that reproduces heavy tails, volatility clustering and rough volatility. Our preferred historical specification combines continuous log-price diffusion, rough stochastic volatility, calendar seasonality and a fixed variance-level shift in January 2022. Five latent coefficients govern variance level, volatility amplitude, roughness, persistence and the shift. Independent Brownian motions drive returns and log variance; no latent price jumps are required by this specification. The level shift is retrospective and does not supply a rule for future breaks. We compare models through a common sampling, detector and variance-proxy map. Fitted through 2022, the four-coefficient baseline scores 9.413 on 2023–2024, versus 9.435 for the eight-coefficient refit and 9.189 for the original eight-coefficient reference. Their January 2025–October 2026 scores are 13.154, 13.515 and 13.225. Four therefore does not dominate both eight-coefficient calibrations. Adding the shift lowers pooled later discrepancy to 5.944 from 7.823 for a matched two-era four-coefficient control. Data resampling and alternative weights support this conditional comparison. Two locally stopped six-coefficient candidates differ in training loss by 0.030385; the better has negligible jump variation. Smooth volatility without jumps scores 15.925 later. The shift model still forces zero leverage and overstates later one-percent tail frequency: 0.532% versus 0.122%. Sixteen coefficients attain training discrepancy 0.568, below the symmetric floor of 0.680, but transfer less well. All latent counts exclude 29 calendar nuisance degrees of freedom. Retained threshold-event records admit sharp hidden-risk bounds. An independent Ornstein–Uhlenbeck observation layer fits variation across horizons but fails signed-return covariance restrictions. Exact first-cell integration, antithetic simulation and independent validation assess numerical error and variability. Bounds, budget stops and Monte Carlo uncertainty are disclosed. These retrospective, single-venue comparisons identify a useful specification within the tested family, not a uniquely optimal equation or a prospective forecasting guarantee. Experiments quantify both improvements and failures.