# Titles and Abstracts

Abstracts of talks for the Stochastic Analysis Conference.

## Hinesh Chotai

### An FBSDE approach to pricing in carbon markets

**Abstract**: Motivated by the work of Carmona, R., Delarue, F., Espinosa, G.E. and Touzi, N., ‘Singular forward–backward stochastic differential equations and emissions derivatives’, and related works, we consider a class of models for cap-and-trade schemes for which the solution to the pricing problem arises from the solution of an appropriate forward backward stochastic differential equation (FBSDE). Their main application is in modelling a (single-period) emissions trading system such as the system in force in the European Union (EU). In contrast to the classes of FBSDEs considered in most of the literature, these FBSDEs are significant in two ways: the terminal condition of the backward equation is given by a discontinuous function of the terminal value of the forward equation, and the forward dynamics may not be strongly elliptic, not even in the neighbourhood of the singularities of the terminal condition. After introducing the model and giving its economic meaning, we will review how a solution to the pricing problem can be constructed. Next, we will consider how the pricing problem can be resolved numerically. Finally, if time permits, we will consider the topic of model calibration and also explain a possible extension to the model that would be applicable for multiple trading periods.

Joint work with Dan Crisan, Jean-François Chassagneux, Mirabelle Muûls.

## Konstantinos Dareiotis

### Symmetrization of exterior parabolic problems and probabilistic interpretation

**Abstract**: We prove a comparison theorem for the ``spatial mass" of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced by F. Brock and A. Solynin. This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved by Y. Peres and P. Sousi.

## Sandy Davie

### CLT bounds in Vaserstein metrics

**Abstract**: We describe an asymptotic expansion in negative powers of n for the Vaserstein 2-distance between the sum of n i.i.d. random vectors and a normal approximation, with an application to a monotonicity question for this distance.

## László Gerencsér

### Epileptic seizure detection

**Abstract**:

Epilepsy is a common disease of the nervous system affecting cca 1% of the population. While epileptic patients may have excellent general health indicators, the unpredictability of seizures effects their quality of life signifcantly. Epilepsy can be controlled with drugs for some patients. For drug-resistant patients, however, a novel therapy is increasingly gaining ground, in which an implant is placed into the patients body monitoring the patients EEG signals, recorded by a portable device, and stimulating the brain prior to the onset of a seizure, preventing its development. Inevitably, part of this complex technology is an appropriate mathematical tool for the analysis of the EEG signals in order to detect achange in its dynamics prior to the onset of seizure.

In this talk we present an approach, borrowed from seismology, in which we first reduce the EEG signal to a point process, loosely speaking indicating events inside the brain, and then focus on the modelling and analysisi of these processes. Thus we arrive at the problem of statistical analysis of so-called self-exiting point processes (Hawkes processes), of which basic properties will be presented, together with methods of simulation, estimation and change detection.

The talk is based on joint work with Loránd Erőss, MD and Dániel Fabó, MD of the National Institute of Clinical Neurosciences, our joint PhD student György Perczel, MD, and his consultant, Zsuzsanna Vágó of the Pázmány Péter Catholic University, Faculty of Information Technology and Bionics.

**References**:

[1] A. G. Hawkes, "Spectra of some self-exciting and mutually exciting point processes", Biometrika, 58:83{90, 1971.

[2] T. Ozaki, "Maximum likelihood estimation of Hawkes self-exciting point processes," Ann. Inst. Stat. Math., vol. 31, no. 1, pp. 145{155, 1979.

[3] J. Moller and J. G. Rasmussen, "Perfect simulation of Hawkes processes," Adv. Appl. Probab., 37(3):629{646, 2005.sic properties will be presented, together with methods of simulation, estimation and change detection.

## Máté Gerencsér

### Boundary value problems via regularity structures

**Abstract**: We discuss the treatment of boundary conditions such as Dirichlet or Neumann for singular SPDEs, within the context of regularity structures. Appropriately treating the arising blow-ups at the various boundaries, the general techniques can be extended to treat terms of regularity greater than -1. Below this threshold new problems arise with the reconstruction operator, the treatment of which we also outline.

## Eric Hall

### Uncertainty quantification for generalized Langevin dynamics

**Abstract **:We present efficient finite difference estimators for the computation of goal-oriented sensitivity indices that apply to a large class of stochastic differential equations with a particular emphasis on generalized Langevin dynamics, a model of diffusive particle dynamics in viscoelastic media. These estimators, which are easy to implement, are formed by coupling the perturbed and nominal dynamics appearing in the finite difference through a common random driving noise.

## Nicolai Krylov

### SPDEs driven by the Poisson process and basic Shauder and Sobolev-space estimates in the theory of parabolic equations

**Abstract**: We show how knowing Shauder and Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the SAME constants as in the case of one-dimensional heat equation. The method is based on using the Poisson stochastic process. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results.

Joint work with E. Priola.

## James-Michael Leahy

### On Degenerate Stochastic Integro-Differential Equations

**Abstract**: TBA

## Chaman Kumar

### Explicit Euler-type scheme of SDE with random coefficient driven by Levy noise: the case of super-linear diffusion coefficient

**Abstract**: We propose an explicit Euler-type scheme for SDE with random coefficient driven by Levy noise when the drift and the diffusion coefficients can grow super-linearly.

Joint work with S. Sabanis.

## Terry Lyons

### From Hopf algebras to machine learning via rough paths

**Abstract**: Rough path theory aims to build an effective calculus that can model the interactions between complex oscillatory (rough) evolving systems. At its mathematical foundations, it is a combination of analysis blended with algebra that goes back to LC Young, and to KT Chen. Key to the theory is the essential need to incorporate additional non-commutative structure into areas of mathematics we thought were stable. At its high points, there are the regularity structures of Martin Hairer that allow robust meaning to be given to numerous core nonlinear stochastic pdes describing evolving interfaces in physics.
Classic results, by Clark, Cameron and Dickinson, demonstrate that a nonlinear approach to the data is essential. Rough path theory lives up to this challenge and can be viewed as providing fundamentally more efficient ways of approximately describing complex data; approaches that, after penetrating the basic ideas, are computationally tractable and lead to new scalable ways to regress, classify, and learn functional relationships from data. One non-mathematical application that is already striking is the use of signatures on a daily basis in the online recognition of Chinese Handwriting on mobile phones.

## Annie Millet

### Soliton dynamics of the mKdV / gKdV equation subject to a random perturbation

**Abstract**: We study random perturbations of solitary solutions to the modified (resp. generalized) KdV equation in dimension 2 (resp. 3). The amplitude of the perturbation is small; we study the orbital stability of solutions as well as exit times from a tube around the solitary solution.

This is a joint work with Svetlana Roudenko.

## Etienne Pardoux

### Random evolution of a population in a changing environment

**Abstract**: We consider a model of the evolution of a population, whose fitness, in the absence of mutations, degradates continuously, due to a constant modification of the ecological conditions (e.g. global warming). We superimpose mutations, which arise according to a Poisson random measure, and get fixed according to a probability which depends upon how much the proposed mutation will improve the fitness. We neglect the time taken for fixation of mutations, and assume that the population is constantly monomorphic. This leads us to consider an SDE driven by a Poisson Point Process, either in dimension 1 or in dimension 2. We study the large time behaviour of the Markov solution of that equation (i.e. its transience/recurrence property). We also study the asymptotic of small/frequent mutations.

This is joint work with Elma Nassar and Mochael Kopp.

## Janos Pintz

### Are Primes Randomly Distributed?

**Abstract**: In the 1930's Cramer created a probabilistic model for primes. He applied his model to express a very deep conjecture about the difference of primes. The general belief was for a period of 50 years that the model reflects the true behaviour of primes when applied to proper problems. It was a great surprise when Helmut Maier discovered in 1985 that the model gives wrong predictions for the distribution of primes in short intervals. We describe how this contradiction could be avoided by a suitable correction of Cramer's model. We present a new contradiction and show that this is present in essentially all Cramer type models. Additionally we report about a new striking discovery of Lemke Oliver and Soundararajan about the unexpected distribution of consecutive primes in arithmetic progressions.

## Michael Röckner

### Global solutions to random 3D vorticity equations for small initial data

**Abstract**: One proves the existence and uniqueness in (L^{p} (R^{3} ))^{3} , 3/2 < p < 2, of a global mild solution to random vorticity equations associated to stochastic 3D Navier-Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in (L^{3} ∩ L^{3p/(4p-6)})^{3} with respect to the time variable. Furthermore, we obtain the pathwise continuous dependence of solutions with respect to the initial data.

This is joint work with Viorel Barbu.

## Andrea Sofia Meireles Rodrigues

### Reference Dependence and Market Participation

**Abstract**: This paper finds optimal portfolios in a one-period model with a safe and a risky asset, with reference-dependent preferences in the sense of Kőszegi and Rabin and piecewise linear gain-loss utility. If the return of the risky asset is highly dispersed relative to its potential gains, two personal equilibria arise, with one of them including risky investments, and the other one only safe holdings. In the same circumstances, the risky personal equilibrium entails market participation that decreases with loss aversion and gain-loss sensitivity, whereas the preferred personal equilibrium is sensitive to market and preference parameters. Relevant market parameters inthe model are not the expected return and standard deviation, but the ratio of expected gains to losses, and the Gini index of the return.