Dr. Cody Hyndman 
Associate Professor and Chair

Department of Mathematics and Statistics

Concordia University

Thursday 22nd November, 2018
4:00pm in MC 5479
​refreshments at 3:45pm

Arbitrage-free regularization, geometric learning, and non-Euclidean filtering in finance

An important concept in mathematical and computational finance is that models should not allow an agent to create (unbounded) profits with zero risk, as any such opportunities should vanish in efficient markets.  This principle is called the "no-free lunch" or the "no-arbitrage" property.  In this talk we present a general overview of some new machine learning algorithms and modeling frameworks that respect the "arbitrage-free" constraint when applied to real financial data. By incorporating non-Euclidean geometry, which can be used to characterize the absence of arbitrage in many financial models, into these algorithms we are able to provide more accurate forecasting and estimation procedures for a wide class of financial models. We first consider the non-Euclidean upgrading (NEU) meta-algorithm which allows us to develop improved versions of standard tools from statistics and machine learning, such as principle component analysis (PCA) and ordinary least squares (OLS), that are commonly used on financial data.  Second, a computationally efficient characterization of conditional expectation for non-Euclidean data is developed to improve upon standard filtering techniques, such as the Kalman filter, and is used to forecast efficient portfolios.  Finally, we consider a flexible modeling framework and a stochastic learning methodology for incorporating arbitrage-free features into many asset price models by utilizing models that may allow for arbitrage, such as the popular Nelson-Siegel factor models, and minimally deforming them into arbitrage-free models.  Numerical results will be presented to illustrate the effectiveness of these new computational methods in finance.

(Based on joint work with Anastasis Kratsios, ETH Zurich)

Dr. Mauro Maggioni
Bloomberg Distinguished Professor, Johns Hopkins University

October 25, 2018
4:00pm in DC 1302
​refreshments at 3:45pm

Learning and Geometry for Stochastic Dynamical Systems in High Dimensions

We discuss geometry-based statistical learning techniques for performing model reduction and modeling of certain classes of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of a system, e.g. from molecular dynamics, and we estimate, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of then system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain, via eigenfunctions of an empirical Fokker-Planck equation (constructed from data), reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas. 

In the second setting we only have access to a (large number of expensive) simulators that can return short paths of the stochastic system, and introduce a statistical learning framework for estimating local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic diffusion-like equations.

Dr. Larry Smith
Economics, University of Waterloo

April 2, 2018
3:30pm in MC 6460
​refreshments at 3:15pm

Computational Mathematics and Its Billion Dollar Problems

Some of the world's most important commercial problems require the use of computational mathematics. Using research from the Problem Lab, we analyse the challenges of these increasingly urgent problems and the reasons they remain fundamentally unsolved. However, there are several promising avenues of investigation. Over the coming terms, there will be opportunities to compete for funding from the Problem Lab to work on these problems.

Dr. Jean-Christophe Nave
Mathematics and Statistics, McGill University

February 6, 2018
4:00pm in MC 6460
​refreshments at 3:45pm