In semimartingale optimal transport problem, the functional to be minimized can be considered as a “stochastic action”, which is the expectationof a “stochastic Lagrangian” in terms of differential semimartingale characteristics. Therefore it would be natural to apply variational calculus approach to characterize the minimizers. R. Lassalle and A.B. Cruzeiro have used this approach to establish a stochastic Euler-Lagrangian condition for semimartingale optimal transport by perturbing the drift terms. Motivated by their work, we want to perform the same type of calculus for martingale optimal transport problem. In particular, instead of only considering perturbations in the drift terms, we try to find a nice variational family for volatility,and then obtain the stochastic Euler-Lagrangian condition for martingale laws. In the first part of this talk we will mention some basic results regarding the existence of minimizers in semimartingale optimal transport problem. In the second part, we will introduce Lassalle and Cruzeiro’s  work, and give a simple example related to this topic, where the variational family is induced by time-changes; and then we will introduce some potential problems that are needed to be solved.

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger,  $L^{2+\epsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.  

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

We consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently, as shown by the numerical study and some theoretical results presented in this talk (based on joint work with Hans Bühler, Ben Wood and Josef Teichmann).

Solutions to Rough Differential Equations (RDE) may be constructed by several means. Beyond the fixed point argument, several approaches rely on using approximations of solutions over short times (Davie, Friz & Victoir, Bailleul, ...). In this talk, we present a generic, unifying framework to consider approximations of flows, called almost flows, and flows through the non-linear sewing lemma. This framework unifies the approaches mentioned above and decouples the analytical part from the algebraic part (manipulation of iterated integrals) when studying RDE. Beyond this, flows are objects with their own properties.New results, such as existence of measurable flows when several solutions of the corresponding RDE exist, will also be presented.

From a joint work with Antoine Brault (U. Toulouse III, France).

In the context of randomly fluctuating surfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang and the Edwards-Wilkinson. Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively. Starting from a modification of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account. We will describe how it arises, briefly discuss its connections to KPZ and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. 

The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic).  Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.

In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1<a<2, and prove convergence to an a-stable Levy process.  This includes to the best of our knowledge the first natural examples where the M_2 Skorokhod topology is the appropriate one.



 

I will consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. A PDE (free boundary problem) approach is used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths.  An Eulerian---mass flow---formulation of the problem is introduced. Its dual is given by Hamilton-Jacobi-Bellman type variational inequalities.  Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglb\"ock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

SLE curves are an important family of random curves in the plane. They share many similarites with solutions of SDE (in particular, with Brownian motion). Any quesion asked for the latter can be asked for the former. Inspired by that, Yizheng Yuan and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea.