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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.
To regain meaningful control over generated volatility the authors break the independence between the data and A by adding a variational volatility encoder qψ(A|x0) that predicts the latent volatility from the data. Crucially, the encoder must be trained with the full stochastic‑volatility likelihood term (the log‑determinant of the noise covariance) that a simplified denoising loss normally drops. With that objective, the encoder becomes an effective volatility filter: interventions on A now steer generation quantitatively, the same ×8 intervention scales the envelope by 3.07 (versus a “naive” scaling of 2.83), and the encoder recovers latent volatility (per‑path correlation 0.57, pooled 0.76) while learning a prior memory that moves toward the true persistence. The authors note costs and limits: the coupled model’s unconditional kurtosis actually falls below the Gaussian baseline, so fidelity to heavy tails and controllability have so far been demonstrated separately, not together.
Why this matters: many recent heavy‑tailed diffusion proposals replaced Gaussian noise with mixtures but ignored temporal dependence in the mixture. Volatility clustering is a central empirical feature of financial returns; DSPM gives a principled, analytically calibrated way to add that persistence into diffusion models, and it clarifies when the mixing mechanism will or will not affect generated data. Important caveats are emphasized in the paper: the invariance result applies in the exact‑denoiser limit and to the conditioned training regime, the variational encoder needs the correct likelihood term to work (otherwise it collapses or leaks), and some empirical tradeoffs (for example between tail fidelity and control) remain. The authors present their derivations and experiments as a working draft and reproduce numbers from accompanying scripts.