Entire sig-jump-diffusions, their expected signature and applications to score-based generative modeling

We develop score-based generative models for time series and standard finite-dimensional data, building on the analytical computation of the expected signature of a new class of jump-diffusions, called entire sig-jump-diffusions. These are semimartingales whose characteristics are given by linear combinations of potentially infinitely many components of their own signature. We show that their expected signature can be computed by solving linear ODEs in the extended tensor algebra, providing a tractable foundation for learning procedures. Specifically, the learning of a generalized score function for jump diffusions, a key ingredient of our generative modeling framework, can be reduced to solving linear ODEs and a deterministic optimization problem. We further prove that polynomial jump-diffusions can be embedded into this framework, and that their expected signature reduces to a finite-dimensional system of linear ODEs. Building on this result, we develop an alternative approach for finite-dimensional data, based on inverse signature matching}, a method that learns the score by matching the expected signature of the reverse process with that of the inverse forward signature.