A Benamou–Brenier formulation of martingale optimal transport

Abstract

We introduce a Benamou-Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker-Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous medium equations is partially investigated.

Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives

“…The semimartingale optimal transport problem with constraints on marginals at given times has been studied by Tan and Touzi in [23], extending the work of Mikami and Thieullen [20]. Other related works include [24,18]. The main goal of our study is to extend this work by considering a much wider range of constraints.…”

Calibration of local‐stochastic volatility models by optimal transport

“…This section is devoted to establishing the duality by closely following Loeper (2006, Section 3.2) (see also Brenier, 1999;Huesmann & Trevisan, 2019).…”

“…Adopting the terminology of Huesmann and Trevisan (2019), we say the triple (𝑟, 𝑎, 𝑏) is represented by (𝜙, 𝜆) if it satisfies…”

“…In Brenier (1999) and Loeper (2006), the dual formulation of the time-continuous optimal transport has been formally expressed and generalized as an application of the Fenchel-Rockafellar theorem (see e.g., Villani, 2003, Theorem 1.9). This method has been also applied in Huesmann and Trevisan (2019) to study a time-continuous formulation of the martingale optimal transport. Recently, the problem has been extended to transport by semi-martingales.…”

Weak Optimal Entropy Transport Problems

“…In this case, for every µ 1 , µ 2 ∈ P(X) the problem (1.2) will become the martingale optimal transport problem. It was introduced first for the case X = R by Beiglböck, Henry-Labordère and Penkner [7] and since then it has been studied intensively [5,6,8,17,22]. Now we introduce our (MOET) problems.…”