A transport view on stochastics and PDEs

In this project we want to systematically study the connection between mass transport theory on the one hand, and stochastic aspects, notably martingale theory, on the other hand. This theme is motivated by applications in Mathematical Finance, pertaining to a robust approach of pricing and hedging exotic options when prices of plain vanilla options are given. One is naturally led to considering a mass transport form a measure 1 to a measure 2 on Rd+ via a martingale. Here the measure 1 (resp. 2 ) models the law of d stock prices at time t1 > 0 (resp. t2 > t1 ) and 2 dominates 1 in convex order. We want to investigate further this topic and develop connections with other applications in PDEs. The main tool is a duality theory based on an application of the minmax theorem which, in turn, is a consequence of the Hahn-Banach theorem. The duality theory opens unexpected insights via the analysis of the dual optimizer which may be viewed as an infinite-dimensional analogue of the notion of a Lagrange multiplier. Our final goal is to derive from the analysis of the dual problem pointwise estimates (instead of estimates in the mean) in several problems, e.g., entropy dissipation in the context of Bakry-Emery. 1

The present project studied in a systematic way the stochastic aspects of diffusions arising in many applications, ranging from statistical mechanics to mathematical finance. The central theme is the "trajectorial" or "pathwise" approach. Given is a stochastic evolution such as a physical system of particles or a number of stocks in a financial market. The randomness is driven in both cases by a Brownian motion. In a landmark paper Jordan, Kinderlehrer and Otto (1998) related the classical theme of entropy dissipation, which goes back to the work of Boltzmann at the end of the nineteenth century, with the theory of optimal transport. In this way they interpreted the heat flow of a physical system as a gradient flow of the entropy functional. The key ingredient is the consideration of the quadratic Wasserstein distance on the pace of probability measures on Euclidean space. In the project at hand we pursued a trajectorial point of view. We consider the relevant quantities such as the entropy, the Fisher information etc. not only as real numbers, but rather as random variables evolving along (almost) every path of the system. We use stochastic calculus, in particular Ito's formula, to deduce their evolution in time. The classical results then follow as simple consequences of these more refined results, pertaining to (almost) each trajectory, by simply taking expectations, i.e., forming averages of the random variables. This approach opens new perspectives and will lead to further applications in various fields, in particular in Mathematical Finance where the PI has already contributed several important results.