The first part of the talk concerns the mathematical modelling of trading in financial markets, with an overview of various ways to incorporate market friction. We then focus on a discrete-time model with transient price impact. Trading widens the bid–ask spread, while market resilience gradually reduces this effect. We present an existence result for an expected-utility-maximizing investment strategy, addressing the difficulty that the set of attainable terminal wealths can be nonconvex. In the second part, we explore the choice of forward process in diffusion-based generative models. Existing theoretical results show that the standard Ornstein–Uhlenbeck process is difficult to improve upon in terms of the time needed to approach equilibrium, within a class of processes whose drift grows at most linearly. The latter restriction leaves open the possibility of improvements through superlinear drift. We discuss ongoing work on superlinear Langevin diffusions, for which preliminary experiments suggest improved generative performance over the Ornstein–Uhlenbeck baseline.