Stochastic Analysis & Mathematical Finance Seminars

Brenier’s theorem and its Benamou-Brenier variant play a pivotal role
in optimal transport theory. In the context of martingale transport
there is a perfect analogue, termed stretched Brownian motion. We
show that under a natural irreducibility condition this leads to the
notion of Bass martingales.
For given probability measures µ and ν on Rn in convex order, the
Bass martingale is induced by a probability measure α. It is the min-
imizer of a convex functional, called the Bass functional. This implies
that α can be found via gradient descent. We compare our approach
to the martingale Sinkhorn algorithm introduced in dimension one by
Conze and Henry-Labordere.

Mathematical modelling and stochastic optimization are often based on the separation of two stages: At the first stage, a model is selected out of a family of plausible models and at the second stage, a policy is chosen that optimizes an underlying objective as if the chosen model were correct. In this talk, I will introduce a new approach which, rather than completely isolating the two stages, interlinks them dynamically. I will first introduce the notion of “consequential performance” of each  model and, in turn, propose a “consequentialist criterion for model selection” based on the expected utility of consequential performances. I will apply the approach to continuous-time portfolio selection and derive a key system of coupled PDEs and solve it for representative cases. I will, also, discuss the connection of the new approach with the popular methods of robust control and of unbiased estimators.   This is joint work with M. Strub (U. of Warwick)

I will present problems a stochastic variant of the classic optimal transport problem as well as a large deviation question for a mean field system of interacting particles. We shall see that those problems can be analyzed by means of a Hamilton-Jacobi equation on the space of probability measures. I will then present the main challenge on such equations as well as the current known techniques to address them. In particular, I will show how the notion of relaxed controls in this setting naturally solve an important difficulty, while being clearly interpretable in terms of geometry on the space of probability measures.

We consider particles following a diffusion process in a multi-well potential and attracted by their barycenter (corresponding to the particle approximation of the Wasserstein flow of a suitable free energy). It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Exactly at the critical temperature below which multiple equilibria appear, the situation is somewhat degenerate and we can get uniform in N convergence estimates, but polynomial instead of exponential.