In this talk, we will establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-H\"older-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds. This is a joint work with Gordan Zitkovic.

We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod (1964), Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in the pair (u,Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our result requires a non-explosion condition which restrict to "small maturity" or "small nonlinearity" of the PDE. Our main ingredient is the automatic differentiation technique as in Henry Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation.

Mining massive amounts of sequentially ordered data and inferring structural properties is nowadays a standard task (in finance, etc). I will present some results that combine and extend ideas from rough paths and machine learning that allow to give a general non-parametric approach with strong theoretical guarantees. Joint works with F. Kiraly and T. Lyons.

Wei Title: Adaptive timestep Methods for non-globally Lipschitz SDEs

Wei Abstract: Explicit Euler and Milstein methods are two common ways to simulate the numerical solutions of
SDEs for its computability and implementability, but they require global Lipschitz continuity on both
drift and diffusion coefficients. By assuming the boundedness of the p-th moments of exact solution
and numerical solution, strong convergence of the Euler-type schemes for locally Lipschitz drift has been
proved in [HMS02], including the implicit Euler method and the semi-implicit Euler method. However,
except for some special cases, implicit-type Euler method requires additional computational cost, which
is very inefficient in practice. Explicit Euler method then is shown to be divergent in [HJK11] for non-
Lipschitz drift. Explicit tamed Euler method proposed in [HJK + 12], shows the strong convergence for the
one-sided Lipschitz condition with at most polynomial growth and it is also extended to tamed Milstein
method in [WG13]. In this paper, we propose a new adaptive timestep Euler method, which shows the
strong convergence under locally Lipschitz drift and gains the standard convergence order under one-sided
Lipschitz condition with at most polynomial growth. Numerical experiments also demonstrate a better
performance of our scheme, especially for large initial value and high dimensions, by comparing the mean
square error with respect to the runtime. In addition, we extend this adaptive scheme to Milstein method
and get a higher order strong convergence with commutative noise.

Alexander Title: Functionally-generated portfolios and optimal transport

Alexander Abstract: I will showcase some ongoing research, in which I try to make links between the class of functionally-generated portfolios from Stochastic Portfolio Theory, and certain optimal transport problems.

In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.
This is joint work with Bob Fernholz and Ioannis Karatzas

We will discuss families of Pell's equation in polynomials 
with one complex parameter. In particular the relation between 
the generic equation and its specializations. Our emphasis will
be on families with a triple zero. Then additive extensions enter 
the picture.