Columbia-NYU Financial Engineering Colloquium: René Carmona & Walter Schachermayer
This event is free, but registration is required.
René Carmona (Princeton University)
Title
Optimal Control of Conditional Processes: Old and New.
Abstract
In this talk, we consider the conditional control problem introduced by P.L. Lions in his lectures at the College de France in November 2016. As originally stated, the problem does not fit in the usual categories of control problems considered in the literature, so its solution requires new ideas, if not new technology. In his lectures, Lions emphasized some of the major differences with the analysis of classical stochastic optimal control problems and in so doing, raised the question of the possible differences between the value functions resulting from optimization over the class of Markovian controls as opposed to the general family of open loop controls. While the equality of these values is accepted as a "folk theorem" in the classical theory of stochastic control, optimizing an objective function whose values strongly depend upon the past history of the controlled trajectories of the system is a strong argument in favor of differences between the optimization results over these two different classes of control processes. The goal of the talk is to elucidate this quandary and provide responses to Lions' original conjecture, both in the case of “soft killing” (R.C. - Lauriere - Lions, Illinois Journal of Math) and in the case of hard killing (R.C. - Lacker, arxiv). We shall also present a new form of Fokker-Planck-Kolmogorov equation for the evolution of the conditional distributions, and discuss the challenges posed by this non-local, non-linear PDE in Wasserstein’s space.
Walter Schachermayer (University of Vienna)
Title
Bass Martingales and Local Volatility
Abstract
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 $\mu$ and $\nu$ on ${\mathbb R}^n$ in convex order, the Bass martingale is induced by a unique probability measure $\alpha$. It is the minimizer of a convex functional, called the Bass functional. This implies that $\alpha$ can be found via gradient descent. We compare our approach to the martingale Sinkhorn algorithm introduced in dimension one by Conze and Henry-Labordère.