We consider an optimal transport problem driven by a Lévy process with long-range jumps, aiming to minimize a cost functional that depends on the stopping time of the process required to transition between an initial and target measure. When the target measure is fixed, this is commonly referred to as the Optimal Skorokhod Embedding problem in the Mathematical Finance literature. In our case, we will allow the target measure to vary as part of the optimization while imposing an upper bound constraint on it.

We will discuss how this problem is linked to the non-local Stefan problem, which describes the phase transition between water and ice with anomalous diffusion. We will focus on the new results obtained due to the probabilistic approach we take, and will discuss open questions. This is joint work with Inwon Kim, Young-Heon Kim, and Kyeongsik Nam.