ETH Zuerich

We present several instances of applications of machine
learning technologies in mathematical Finance including pricing,
hedging, calibration and filtering problems. We try to show that
regularity theory of the involved equations plays a crucial role
in designing such algorithms.

(based on joint works with Hans Buehler, Christa Cuchiero, Lukas
Gonon, Wahid Khosrawi-Sardroudi, Ben Wood)

In this lecture I intend to review a few selected recent results on numerical approximations for high-dimensional nonlinear parabolic partial differential equations (PDEs), nonlinear stochastic ordinary differential equations (SDEs), and high-dimensional nonlinear forward-backward stochastic ordinary differential equations (FBSDEs). Such equations are key ingredients in a number of pricing models that are day after day used in the financial engineering industry to estimate prices of financial derivatives. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on the Heston model, as well as on nonlinear pricing models for financial derivatives. We illustrate our results by several numerical simulations and we also calibrate some of the considered derivative pricing models to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.

Numbers like the special values of the gamma and the Bessel functions or the Euler-Mascheroni constant are not expected to be periods in the usual sense of algebraic geometry. However, they can be regarded as coefficients of the comparison isomorphism between two cohomology theories associated to pairs consisting of an algebraic variety and a regular function: the de Rham cohomology of a connection with irregular singularities, and the so-called “rapid decay cohomology”. Following ideas of Kontsevich and Nori, I will explain how this point of view allows one to construct a Tannakian category of exponential motives over a subfield of the complex numbers. The upshot is that one can attach to exponential periods a Galois group that conjecturally governs all algebraic relations between them. Classical results and conjectures in transcendence theory may be reinterpreted in this way. No prior knowledge of motives will be assumed, and I will focus on examples rather than on the more abstract aspects of the theory. This is a joint work with P. Jossen (ETH Zürich).

We will discuss the concept of $\ell^2$-stability of a group and show some of its rigidity consequences.  We provide moreover some very concrete examples of lattices in product of trees that have many interesting properties, $\ell^2$-stability being only one of them.

Starting with seminal work by Masur-Minsky, a lot of machinery has been
developed to study the geometry of Mapping Class Groups, and this has
lead, for example, to the proof of quasi-isometric rigidity results.
Parts of this machinery include hyperbolicity of the curve complex, the
distance formula and hierarchy paths.
As it turns out, all this can be transposed to the context of CAT(0)
cube complexes. I will explain some of the key parts of the machinery
and then I will discuss results about quasi-Lipschitz maps from
Euclidean spaces and nilpotent Lie groups into "spaces with a distance
formula".
Joint with Jason Behrstock and Mark Hagen.

After a short survey of the notion of level of distribution for
arithmetic functions, and its importance in analytic number theory, we
will explain how our recent studies of twists of Fourier coefficients of
modular forms (and especially Eisenstein series) by "trace functions"
lead to an improvement of the results of Friedlander-Iwaniec and
Heath-Brown for the ternary divisor function in arithmetic progressions
to prime moduli.

This is joint work with É. Fouvry and Ph. Michel.