Gabriele Bocchi's TR-SBTS Generates Time Series with Hierarchical Volatility

Geometric methods for time series generation get a new twist with Gabriele Bocchi’s Triangular-Reference Schrödinger Bridges for Time Series (TR-SBTS). The key innovation: swapping the standard Brownian bridge reference for an intervalwise frozen, possibly degenerate diffusion reference that is triangular across a hierarchy of latent volatility levels. This single entropy projection on the augmented state space imposes variational constraints jointly across time and latent levels, unfolded hierarchically by the disintegration of relative entropy. The core Schrödinger bridge property is preserved: the entropy minimiser remains the h-transform of the reference, and on each frozen interval the optimal dynamics admit a logarithmic-gradient drift formula valid even when the covariance is rank-deficient. Bocchi also proves stability of the frozen approximation and convergence of regularised kernel estimators.

TR-SBTS’s practical realisation relies on a finite-dimensional conditioning map assembled from three complementary reductions of the past: a block PCR summary, a reference-aware Mahalanobis kernel on past increments induced by the runtime frozen covariance cumulants, and a past-window WLS drift regressor under the same reference metric. A coupled state-covariance bridge step lets each latent level produce a dynamic reference for the level above, summarised by a covariance descriptor. Numerical experiments validate the construction. This work extends the SBTS framework and provides a mathematically rigorous path for generating time series with hierarchical latent structure, with potential applications in finance, climate modeling, and synthetic data generation.