$q$-Bass martingales

An intriguing question in martingale optimal transport is to characterize the martingale with prescribed initial and terminal marginals whose transition kernel is as Gaussian as possible. In this work we address an extension of this question, in which the role of the Gaussian distribution is replaced by an arbitrary reference measure $q$. Our first main result is a dual formulation of the corresponding martingale optimization problem in terms of convex functions. In the well-studied case when $q$ is Gaussian, the careful analysis of the solution to the above-mentioned optimization problem is a crucial building block in the construction of Bass martingales, i.e., Brownian martingales induced by gradients of convex functions with possibly non-degenerate starting laws. In our second main result we extend this concept beyond the Gaussian case by introducing the notion of $q$-Bass martingales in discrete time, and give sufficient conditions for their existence.