Time Series Constrained Sampling via Refining Trajectory-Level Control

Time series generation is central to applications such as finance, energy, and biomedical sensing, where realistic samples must respect partial observations or structural constraints. Existing training-free approaches based on guidance or projection enforce such constraints only locally at each diffusion step, which we show is insufficient to recover the conditional posterior in path space. In this work, we reformulate constrained sampling for time-series diffusion models as a trajectory-level stochastic optimal control problem, and unify guidance- and projection-based methods as instances of local control within this framework. We propose Trajectory-level Time-series Control (TTC), a training-free, inference-time adjoint scheme that parameterizes the control in Fourier and wavelet subspaces aligned with global and localized temporal structure. A single backward pass already moves samples substantially closer to the constraint set at guidance-level cost, additional iterations tighten this further, and a terminal projection supplies exact feasibility. Across four datasets and multiple constraint types, TTC reaches exact constraint satisfaction with the best distributional quality among the compared methods, while running up to an order of magnitude faster than per-step projection.