Constraint-Driven Measure Transformation of Pretrained Diffusion Models

Adapting a pretrained diffusion model to a target known only through expectation constraints requires solving for the tilt that reaches it. We solve for it by minimum relative entropy against the learned prior, giving the weighted Monte Carlo measure, and carry it into the frozen sampler through the Doob $h$-transform. Both are prior work. Because the reference is learned, it can be varied, and the construction becomes an instrument for a question central to derivative pricing: what does a finite set of market constraints determine about a path-dependent price? On Heston paths, 53 constraints span 97% of an Asian option's payoff variance but only 76% of a barrier's, and that unspanned variance sets, to within 2.4%, how fast the admissible price range opens with relative-entropy budget. On a real AAPL option surface, every prior we test (learned, parametric and Black–Scholes alike) misprices around a scheduled earnings date in the same direction, and the pipeline recovers the known earnings-jump signature from the quotes. Carrying the tilt into the sampler, which the identification results do not require, works only at mild tilts. The learned correction is accurate as a function, but the sampler does not realise the measure it implies.