Title of the speech: Dynamic universal approximation and optimal control for non-Markovian systems via signature stochastic differential equations
Abstract:
Many applications in generative time series modeling, particularly in finance, require stochastic dynamics that are genuinely path-dependent and non-Markovian. Classical Markovian state-space models are often too restrictive to capture memory effects, delayed responses, volatility feedback, and other features generated by the past trajectory of the system. From a numerical perspective, the most natural route is to lift path dependence to an enlarged state space, thereby obtaining a Markovian approximation of the original dynamics. Among such lifts, signature stochastic differential equations (SDEs) provide a canonical and model-agnostic choice, building on the strong algebraic and approximation-theoretic properties of path signatures, which form a universal and non-parametric feature set for paths. We explain how this leads to dynamic universal approximation results for generic non-Markovian SDEs. We also show how the resulting finite-dimensional signature SDEs can be used in optimal control problems with path-dependent dynamics and objectives, reducing, for instance, the infinite-dimensional Hamilton-Jacobi-Bellman PDE to a much simpler Riccati ODE.