Signatures in stochastic control: open loop and beyond

We explore recent advances in stochastic optimal control beyond the Markovian setting, using path signatures in several aspects. Beyond the recent open-loop parameterization approach from [Bayer et al., "Stochastic Control with Signatures," '24] and its accompanying Monte Carlo method, we discuss recent progress in treating the closed-loop and path-dependent framework. In another direction, we explore a duality framework where signatures emerge through a functional Taylor expansion of the pathwise penalty term. Together, these methods provide new insights into both the primal and dual formulations of stochastic control, with applications in non-Markovian Mathematical Finance modeling.