Mathematical Finance Seminar – Spring 2021

Schedule for Spring 2021

Seminars are on Thursdays
Time: 4:10pm – 5:25pm
Attention: All talks are available online, via Zoom. Select talks take place in hybrid mode. In-person participation is only available to Columbia affiliates with building access.

Join URL:

Meeting ID: 934 6125 7216

Passcode: Bachelier

Organizers: Ioannis Karatzas, Marcel Nutz, Philip Protter, Xiaofei Shi, Johannes Wiesel

MAFN Seminar Archive

Xunyu Zhou (Columbia)

“Curse of optimality, and how do we break it”


We strive to seek optimality, but often find ourselves trapped in bad “optimal” solutions that are either local optimizers, or are too rigid to leave any room for errors, or are simply based on wrong models or erroneously estimated parameters. A way to break this “curse of optimality” is to engage exploration through randomization. Exploration broadens search space, provides flexibility, and facilitates learning via trial and error. We review some of the latest development in this exploratory approach in the stochastic control setting with continuous time and spaces.

Johannes Wiesel (Columbia)
“Data driven robustness and sensitivity analysis”

Abstract: In this talk I consider sensitivity of a generic stochastic optimization problem to model uncertainty, where I take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. I provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. Then I present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, I prove that LASSO leads to parameter shrinkage, propose measures to quantify robustness of neural networks to adversarial examples and compute sensitivities of optimised certainty equivalents in finance. I also propose extensions of this framework to a multiperiod setting. This talk is based on joint work with Daniel Bartl, Samuel Drapeau and Jan Obloj.


*Start Time: 2:30pm

*End Time: 3:30 pm

Joint with the Applied Probability and Risk seminar.

Soumik Pal (University of Washington Seattle)

“A Gibbs measure perspective on Schrodinger bridges and entropy regularized optimal transport.”

Abstract:  Consider the problem of matching two independent sets of N i.i.d. observations from two densities. Such matchings correspond to the set of permutations of N labels. For an arbitrary continuous cost function, the optimal assignment problem looks for that permutation that minimizes the total cost of matching each pair of atoms. The empirical distribution of the matched atoms is known to converge to the solution of the Monge-Kantorovich optimal transport problem.

Suppose instead we take a weighted convex combination of the empirical distribution of every matching, weighted proportional to the exponential of their (negative) total cost. Then the resulting distribution converges to the solution of a variational problem, introduced by Follmer, called the entropy-regularized optimal transport. This weighted combination is a variant of entropy regularization for discrete optimal transport similar to the one due introduced by Cuturi for faster computations. For this variant one can describe limiting Gaussian and non-Gaussian distributions that are useful in statistical estimation. 

As a big picture, we will discuss how discrete optimal transport problems can be analyzed by classical tools such as U-statistics, exchangeability and combinatorics of symmetric functions. This avoids the use of analytical machinery on metric measure spaces that are frequently used in such problems for the quadratic cost but are unavailable outside that of the Wasserstein spaces.



Joint with the Applied Probability and Risk seminar.

Julio Backhoff-Veraguas (Vienna)

“The mean field Schrödinger problem: large deviations and ergodic behaviour.”
Abstract: In the classical Schrödinger problem the aim is to minimize a relative entropy cost over the laws of processes with a prescribed initial and terminal marginals. Via large deviations theory, a solution to the Schrödinger problem approximates the distribution of a large system of independent particles conditioned to have a prescribed initial and terminal configuration. In the first part of this talk I will explain how the Schrödinger problem looks like when instead of independent particles we allow for weakly dependent ones. Specializing the discussion to a diffusion model with mean field interactions, I will illustrate in the second part of this talk how the effect of conditioning at initial and terminal times is exponentially small at intermediate times under precise ergodicity assumptions.
Based on joint work with Conforti, Gentil and Leonard.


Guillaume Carlier (Dauphine)

“A mean field game model for the evolution of cities”

Abstract: In this talk, I will present a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions  and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme. I will present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters. This is a joint work with César Barilla and Jean-Michel Lasry.