Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs

In this paper, we investigate the multi-marginal Schrodinger bridge (MSB) problem whose marginal constraints are marginal distributions of a stochastic differential equation (SDE) with a constant diffusion coefficient, and with time dependent drift term. As the number $m$ of marginal constraints increases, we prove that the solution of the corresponding MSB problem converges to the law of the solution of the SDE at the rate of $O(m^{-1})$, in the sense of KL divergence. Our result extends the work of~\cite{agarwal2024iterated} to the case where the drift of the underlying stochastic process is time-dependent.