Researchers introduce the Denoising Subordinated Probabilistic Model (DSPM), a variant of denoising diffusion models that directly models persistent changes in noise amplitude — the “volatility clock” that creates volatility clustering in financial returns. In plain terms, instead of adding independent Gaussian or heavy-tailed noise to each coordinate, DSPM gives each coordinate a random variance multiplier A and lets those multipliers evolve along the generated data as a simple persistent time series. That chain of variances is driven by tempered-stable jumps, a mathematically convenient way to allow heavy but finite tails in the variance process.

Technically, DSPM makes the mixing vector A a stationary AR(1) chain (an autoregressive process with one lag) whose increments come from a tempered-stable subordinator. Conditionally on a realized A, the usual denoising diffusion (DDPM — denoising diffusion probabilistic model) training and sampling steps work unchanged. The authors derive closed-form formulas that link the chain parameters to two observable features: excess kurtosis (a measure of heavy tails) and the autocorrelation of squared noise (a proxy for volatility clustering). They give an analytic inverse map for calibration: for target excess kurtosis K and lag‑one squared autocorrelation ρ1 the memory parameter φ and tempering parameter θ can be recovered explicitly, and they show a feasibility bound ρ1 < K/(3(K+2)) that the model class must satisfy.

The paper also proves a delimiting theoretical result about what the noise mechanism actually controls in these models. If the denoiser is conditioned on the variance chain A (that is, the model sees the per-coordinate multipliers during training and sampling), then the detailed probability law that generates A becomes a nuisance: in the ideal “exact denoiser” limit, the generated data distribution does not depend on that law, and direct interventions on A have no effect. The authors test this in experiments. Conditioned models reproduce volatility clustering no matter what exact law was used to draw A, and a designed ×8 volatility shock changed the generated amplitude envelope by under 13% (compared to a naive prediction of a much larger change). By contrast, “blind” models that do not receive A simply transmit the noise mechanism directly: the calibrated lag‑one squared‑return autocorrelation appears in samples nearly exactly, while the i.i.d. mixing variant loses clustering almost completely.