We consider the bicausal optimal transport problem between the laws of scalar time-homogeneous stochastic differential equations, and we establish the optimality of the synchronous coupling between these laws. The proof of this result is based on time-discretisation and reveals a novel connection between the synchronous coupling and the celebrated discrete-time Knothe–Rosenblatt rearrangement. We also prove a result on equality of topologies restricted to a certain subset of laws of continuous-time processes. We complement our main results with examples showing how the optimal coupling may change in path-dependent and multidimensional settings.

We introduce an efficient computational framework for solving a class of multimarginal martingale optimal transport problems, which includes many robust pricing problems of large financial interest. Such problems are typically computationally challenging due to the martingale constraint; however, by extending the state space we can identify them with problems that exhibit a certain sequential martingale structure. Our method exploits such structures in combination with entropic regularization, enabling fast computation of optimal solutions and allowing us to solve problems with a large number of marginals. We demonstrate the method by using it for computing robust price bounds for different options, such as lookback options and Asian options.

We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We establish the “classical” results of stochastic control for these problems: specifically, we prove that the value function for the problem can be characterised as the unique solution to the Hamilton–Jacobi–Bellman equation in the sense of viscosity solutions. In order to prove this result, we exploit structural properties of the MVM processes. Our results also include an appropriate version of Itô’s formula for controlled MVMs. We also show how problems of this type arise in a number of applications, including model-independent derivatives pricing, the optimal Skorokhod embedding problem, and two player games with asymmetric information.

We consider an optimal stopping problem where a constraint is placed on the distribution of the stopping time. Reformulating the problem in terms of so-called measure-valued martingales enables us to transform the distributional constraint into an initial condition and view the problem as a stochastic control problem; we establish the corresponding dynamic programming principle. The method offers a systematic approach for solving the problem for general constraints and under weak assumptions on the cost function. In addition, we provide certain continuity results for the value of the problem viewed as a function of its distributional constraint.

We study an inverse investment problem proposed by Black and provide necessary and sufficient conditions for a given function to be an admissible indirect utility function in a log-normal market; we also show how to recover the associated utility function. Similar questions are also addressed starting directly from the initial investment choice. In parallel we study so-called forward investment-consumption criteria with the dynamic property that their volatility component is identically zero. We provide a fully forward characterization of such criteria and use it to construct forward preferences. We also provide explicit formulas for the associated optimal strategies and characterize the class of criteria which may be decomposed into a pure forward investment criterion and an infinite horizon Merton problem.

In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are corner-stones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport problem for given d-dimensional distributions mu, nu in convex order. The unique solution M* = (M-t*)(t is an element of[0,1]) of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the con ditions M-0*similar to mu, M-1*similar to nu. Similar to McCann's displacement-interpolation, M* provides a time-consistent interpolation between mu and nu. For particular choices of the initial and terminal law, M* recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer's theorem. This article is parallel to the work of Huesmann-Trevisan, who consider a related class of problems from a PDE-oriented perspective.