Bridging Schr\"odinger and Bass: A Semimartingale Optimal Transport Problem with Diffusion Control

We study a semimartingale optimal transport problem interpolating between the Schr\"odinger bridge and the stretched Brownian motion associated with the Bass solution of the Skorokhod embedding problem. The cost combines an entropy term on the drift with a quadratic penalization of the diffusion coefficient, leading to a stochastic control problem over drift and volatility. We establish a complete duality theory for this problem, despite the lack of coercivity in the diffusion component. In particular, we prove strong duality and dual attainment, and derive an equivalent reduced dual formulation in terms of a variational problem over terminal potentials. Optimal solutions are characterized by a coupled Schr\"odinger-Bass bridge system, involving a backward heat potential and a transport map given by the gradient of a $\beta$-convex function. This system interpolates between the classical Schr\"odinger system and the Bass martingale transport. Our results furnish a unified framework encompassing entropic and martingale optimal transport, and yield a variational foundation for data-driven diffusion models.