I present recently developed iterated residue formulas for tautological integrals over Hilbert schemes of points on smooth manifolds. Applications include curve and hypersurface counting formulas. Joint work with Andras Szenes.
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)
We discuss two Freidlin-Wentzell large deviation principles for McKean-Vlasov equations (MV-SDEs) in certain path space topologies. The equations have a drift of polynomial growth and an existence/uniqueness result is provided. We apply the Monte-Carlo methods for evaluating expectations of functionals of solutions to MV-SDE with drifts of super-linear growth. We assume that the MV-SDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the "complete measure change" approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the "decoupling" approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step.
We present a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, and bounds on the distribution of a sum of dependent random variables. As an application we focus on the problem of risk aggregation under model uncertainty. The talk is based on joint work with Stephan Eckstein and Mathias Pohl.
High-frequency realized variance approaches offer great promise for
estimating asset prices’ covariation, but encounter difficulties
connected to the Epps effect. This paper models the Epps effect in a
stochastic volatility setting. It adds dependent noise to a factor
representation of prices. The noise both offsets covariation and
describes plausible lags in information transmission. Non-synchronous
trading, another recognized source of the effect, is not required. A
resulting estimator of correlations and betas performs well on LSE
mid-quote data, lending empirical credence to the approach.
Title: Generalized McKean-Vlasov stochastic control problems.
Abstract: I will consider McKean-Vlasov stochastic control problems
where the cost functions and the state dynamics depend upon the joint
distribution of the controlled state and the control process. First, I
will provide a suitable version of the Pontryagin stochastic maximum
principle, showing that, in the present general framework, pointwise
minimization of the Hamiltonian with respect to the control is not a
necessary optimality condition. Then I will take a different
perspective, and present a variational approach to study a weak
formulation of such control problems, thereby establishing a new
connection between those and optimal transport problems on path space.
The talk is based on a joint project with J. Backhoff-Veraguas and R. Carmona.
In this talk we present a framework for discretely compounding
interest rates which is based on the forward price process approach.
This approach has a number of advantages, in particular in the current
market environment. Compared to the classical Libor market models, it
allows in a natural way for negative interest rates and has superb
calibration properties even in the presence of persistently low rates.
Moreover, the measure changes along the tenor structure are simplified
significantly. This property makes it an excellent base for a
post-crisis multiple curve setup. Two variants for multiple curve
constructions will be discussed.
As driving processes we use time-inhomogeneous Lévy processes, which
lead to explicit valuation formulas for various interest rate products
using well-known Fourier transform techniques. Based on these formulas
we present calibration results for the two model variants using market
data for caps with Bachelier implied volatilities.
Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDES. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, and outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.
In this talk I will discuss some recent constructions of blow-up solutions for a Fujita type problem for power related to the critical Sobolev exponent. Both finite type blow-up (of type II) and infinite time blow-up are considered. This research program is in collaboration with C. Cortazar, M. del Pino and J. Wei.