Elisa Nicolato

Simple Smiles For The Mixing Setup. Slides

We derive a closed-form approximation of implied volatility and at-the-money skew. The approximation is simple, transparent, and easy to implement. It is based on a generalized mixing solution of the underlying model. We apply the approximation to the Heston and the 3/2 stochastic volatility models. We also consider Lévy models such as the Variance Gamma model. However, the approximation may be used for a broad class of option pricing models displaying both jumps and stochastic volatility in the underlying dynamics. Finally, we explore two domains of application of the expansion. One as a control variate to speed up Fourier option pricing, and another one as a fast, first-order approach for calibration to at-the-money volatilities and skews.

Franco Flandoli

Short introduction to some topics in Mathematical Oncology. Slides

Some topics at the link between nonlinear PDEs and interacting systems of particles will be selected, motivated by problems of growth and diffusion of cancel cells. The PDEs are similar to Fokker-Planck equations, with reaction terms. The interacting particles aim to include features typical of cells. Several open problems arise. The treated topics have a foundational character; the limitations for realistic applications will be discussed.

Dario Spanò

Bayes and Lancaster at the Chinese Restaurant: statistical uses of the Fleming-Viot process. Slides

The neutral Fleming-Viot process is a well-understood measure-valued diffusion playing a central role in Population Genetics. Much of its tractability is owed to the transparent structure of its transition function, which has (at least) two insightful representations: one in terms of systems of dependent Chinese Restaurant Processes (Polya-type urn sequences), the other, more classically, in terms of so-called Lancaster probabilities (kernels with orthogonal polynomial eigenfunctions). Kingman’s coalescent describes the link between the two interpretations. I will illustrate some Bayesian-flavoured statistical uses of the Fleming-Viot, and the power of its transition function in generating computable non-parametric filters and exact simulation algorithms for diffusion sample paths.

• Andreis Luisa - Università di Padova

McKean-Vlasov limit for interacting systems with simultaneous jumps. Slides

We consider systems of N weakly interacting diffusions with jumps, having the peculiar feature that the jump of one component may induce simultaneous jumps of all others. Models belonging to this class have been proposed for the dynamics of neuronal systems, and their limiting $$N\to\infty$$ behaviour has been studied only for special cases. We prove propagation of chaos and derive the corresponding McKean-Vlasov equation. Unlike existing approaches in neuronal models, we show that classical tools can be profitably adapted to treat this class of systems with a good degree of generality.

• Catellier Rémi - Université de Rennes 1 IRMAR

Averaging along irregular curves and regularization of ODEs. Slides

Paths of some stochastic processes such as fractional Brownian motion have some amazing regularizing properties. It is well known that in order to have uniqueness in differential systems such as \begin{equation*}\mathrm{d} {y}_t=b(y_t)\mathrm{d} t\end{equation*} $$b$$ needs to be quite regular. As soon as the last equation is perturbed by a suitable stochastic process $$w$$, the oscillations of such a process will guarantee that the following system $$y_t = x + \int_0^t b(y_r)\mathrm{d} r + w_t \label{eq:ODE}$$ has a unique solution for really irregular $$b$$. After recalling some basic facts we will show that the study of the following stochastic averaging operator \begin{equation*}T^w_t b(x)=\int_0^t b(x+w_r)\mathrm{d} r\end{equation*} will allow us to solve equation (\ref{eq:ODE}) for $$b$$ on which we have some control on the growth of the Fourier transform. This will allow us to extend such equations when $$b$$ are not functions but distributions. As an application, we will show that the stochastic transport equation driven by fractional Brownian motion with $$H\in(0,1)$$ \begin{equation*}\partial_t u + b.\nabla u+\nabla u.\mathrm{d} B^H_t=0\end{equation*} has a unique solution when $$u_0\in L^{\infty}$$ and $$b$$ is a possibly random $$\alpha$$-Hölder continuous function for $$\alpha$$ large enough. This is a joint work with Massimiliano Gubinelli

• Cipriani Alessandra - WIAS

Thick points for generalized Gaussian fields with different cut-offs. Slides

In this talk we would like to present a result on the Hausdorff dimension of the so-called a-thick points for generalized Gaussian fields with logarithmically diverging variance. Such fields are not functions, but distributions, hence it is useful to approximate them by suitable cut-off random variables. The choice for cut-offs is rather broad, hence we would like to understand under which assumptions the Hausdorff dimension is universal. We will show that the dimension equals $$d-a^2/2$$ as predicted by Kahane's theory of Gaussian multiplicative chaos, and we will try to explain how our result complements the ones obtained by Kahane, Hu-Miller-Peres and Rhodes-Vargas in the context of Gaussian multiplicative chaos and the $$2-d$$ Gaussian Free Field. This is a joint work with Rajat Subhra Hazra.

• Clausel Marianne - Grenoble University

Stein estimation of the intensity of a spatial homogeneous Poisson point process. Slides

In this paper, we revisit  the original ideas of Stein and propose an estimator of the intensity parameter of a homogeneous spatial Poisson point process defined and observed in a bounded window. The procedure is based on a new general integration by parts formula for Poisson point processes. We show that our Stein estimator outperforms the maximum likelihood estimator in terms of mean squared error. In particular, we show that in many practical situations we have a gain larger than 30%.

• Coghi Michele - Scuola Normale Superiore di Pisa

Many Particle Systems and Mean Field Limits. Slides

We consider a system of interacting particles described by stochastic differential equations. Opposite to the usual scheme, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similarly to to the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of inviscid type, opposite to the case when independent noises drive the different particles.

• Dahlqvist Antoine - TU Berlin

Maximal angle of a system of particles on the circle. Slides

We will explain how the moment method allows to study the maximum angle of a system of self-repelling particles on the circle that is associated to a diffusion on unitary matrices.

• Fahrenwaldt Matthias - Leibniz Universität Hannover

Spectral theory of subordinated Brownian motion. Slides

For a class of Bernstein functions we consider the spectrum of the generator of the corresponding subordinated Brownian motion by analyzing the operator zeta function and the heat trace. Our key assumption is the existence of a density for the Levy measure with asymptotic expansion near the origin and rapid decay at infinity. Each of the spectral functions is then determined explicitly in terms of local properties of this density. We employ methods from the theory of classical pseudodifferential operators on Euclidean space. The analysis is highly explicit and fully analytically tractable.

• Gabrielli Nicoletta - ETH Zurich

Lamperti transform for multi-type continuous state and continuous time branching processes. Slides

Based on the theory of multivariate time change for Markov processes, we show how to identify multi-type branching processes as solutions of certain time change equations. More precisely, we are able to reconstruct their paths from the paths of a family of Levy processes properly time–changed. In dimension one, this transformation on the path-space transforming the laws of Levy processes with no negative jumps to the law of branching processes is known as Lamperti transform.

• Ganychenko Iurii - Taras Shevchenko National University of Kyiv

$$L_2$$-rates of approximation of non-smooth integral type functionals in case of additional regularity. Slides

We establish strong $$L_2$$-rates of approximation of non-smooth integral type functionals of Markov processes by integral sums. The case of additional regularity for the argument function is considered. The assumptions on the process are formulated only in the terms of its transition probability density, and therefore are quite flexible vs. existent results.

• Gaviraghi Beatrice - University of Würzburg

An operator splitting method for solving a class of Fokker-Planck equations Slides

The Fokker-Planck equation describes the time evolution of the probability density function of a stochastic process. When the underlying process contains jumps, the evolution of its probability density function is modeled by a partial-integro differ- ential equation. We consider an initial-value problem containing the partial-integro differential Fokker-Planck equation, whose integral part is due to a finite activity jump process. We propose a convergent and stable operator splitting method that guarantees the two required properties of the probability density function, namely its positivity and the conservativeness of the total probability.

• Ghilli Daria - Università di Padova

Large deviations for some fast stochastic volatility models by viscosity methods. Slides

The topic of the talk is to present some recent results about short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.

• Houdebert Pierre - Université Lille 1

Continuum Widom-Rowlinson model. Slides

The Widom-Rowlinson model has been introduced first by Widom and Rowlinson in the early 70' to describe system of two types of ga particles with a hard-core interaction between particles. In my talk I will first start with standard definitions of the spatial Poisson point process. Then I will introduce the Widom-Rowlinson model on a bounded set, where there is no difficulties of definition or existence. Finally in the infinite volume case I will give existence results and some of the classical tools that lead to these results.

• Kosenkova Tetiana - Taras Shevchenko National University of Kyiv

Levy-type processes: characterisation and convergence. Slides

In this talk we give an overview of the results on characterisation of Levy-type processes. Namely, we consider different approaches to the Levy-type processes construction: semigroups methods, martingale characterisation, and construction of the process as a solution to SDE. In addition, we consider convergence schemes, where Levy-type processes appear as limit points.

• Kruse Raphael - TU Berlin

On the weak approximation of solutions to stochastic partial differential equations. Slides

VIn numerical analysis of stochastic partial differential equations (SPDEs) one usually differentiates between the notions of strong and weak convergence. While the first notion ensures a good pathwise approximation of the numerical discretization to the solution of the SPDE, a weakly convergent scheme only gives a good approximation of its law. Strong convergence implies weak convergence and, by a rule of thumb, the order of weak convergence is often up to twice the order of strong convergence. In this talk we propose a new approach to prove the weak order of convergence, which avoids the classical ansatz relying on the associated Kolmogorov’ backward equation. Instead we follow a more direct approach with a Gelfand triple based on a suitable subspace of the Sobolev-Malliavin space. The weak error analysis is then carried out by practically the same techniques as known from the strong error analysis but with a weaker norm. This ansatz is used to prove a weak convergence result for an Euler Galerkin finite element approximation of an SPDE with additive noise. This is joint work with Adam Andersson and Stig Larsson (both Chalmers University of Technology).

• Lie Han Cheng - FU Berlin

An application of Girsanov's theorem to a model of molecular dynamics. Slides

Diffusions on energy landscapes are often used in statistical-mechanical models of chemical reactions, because the simulation of molecular dynamical systems as diffusions yields empirical statistics that agree with the predictions of physics. However, obtaining accurate estimates of statistical properties continues to pose challenges in computational molecular dynamics, many of which involve the problem of sampling rare events. So-called `nonequilibrium methods' aim to avoid these problems by applying a biasing force. We present one such method which uses Girsanov's theorem and some ideas from stochastic control.

• Mukherjee Chiranjib - TU Munich

Brownian occupation measures, compactness and large deviations. Slides

Much studied classical Donsker-Varadhan theory proves a weak large deviation principle for the occupation measures of Brownian motion in the space $$\mathcal{M}_1(\mathbb{R}^d)$$ of probability distributions in $$\mathbb{R}^d$$, $$d\geq1$$. This space, equipped with the weak topology is not compact. In this article, via the notion of shift compactness we compactify its quotient space $$\widetilde{\mathcal{M}}_1(\mathbb{R}^d)$$ of orbits under translations, and prove a strong large deviation principle for the distribution of the empirical measures of Brownian motion in this compact space. Our result overcomes a number of technical hindrances stemming from the lack of a strong large deviation principle of empirical measures of Markov processes on non-compact spaces. (Joint work with S. R. S. Varadhan).

• Peña Helena - Greifswald Universität

Approximation of the invariant measure of an Iterated Function System. Slides

An Iterated Function System (IFS) on $$X= \mathbb{R}^n$$ or $$\mathbb{C}^n$$ consists of mappings $$f_i:X\rightarrow X$$ and probabilities $$p_i$$. An IFS describes a stochastic dynamical system. Its stationary distribution is the invariant measure of the IFS. To an IFS with affine mappings, we associate an operator acting on the space $$C(X)$$ of continuous functions. For the case $$n=1$$ we can compute all its polynomial eigenfunctions. These yield an approximation for the invariant measure. As an example, we consider Bernoulli convolutions.

• Pigato Paolo - Università di Padova

Density estimates for a chain of stochastic differential equations. Slides

We derive some bounds for densities of random variables, starting from the Malliavin and Thalmaier representation formula in terms of the Riesz transform. We apply them to a chain of $$n$$ differential equations of dimension $$d$$, finding Gaussian estimates for the density of the solution in short time under a weak Hörmander condition.

• Riedel Sebastian - TU Berlin

Transporation-cost inequalities and rough paths

Transportation-cost inequalities can be used to characterize the concentration properties of probability measures. In this talk, we will investigate the law of diffusions driven by a multidimensional Gaussian process and we will show that Lyons' rough paths theory is a powerful tool to establish such inequalities on path spaces.

• Rossi Maurizia - Roma Tor Vergata

Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on $$S^d$$. Slides

We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit d-dimensional sphere $$S^d$$, $$d\geq 2$$. All our results are established in the high energy limit, i.e. for eigenfunctions corresponding to growing eigenvalues. More precisely, we provide an asymptotic analysis for the variance of random eigenfunctions, and also establish rates of convergence for various probability metrics for Hermite subordinated processes, arbitrary polynomials of finite order and square integral nonlinear transforms; the latter, for instance, allows to prove a quantitative Central Limit Theorem for the excursion area. Some related issues were already considered in the literature for the 2-dimensional case $$S^2$$, in connection with the analysis and the modeling of the Cosmic Microwave Background; our results are new or improve the existing bounds even for this special case. Proofs are based on the asymptotic analysis of moments of all order for Gegenbauer polynomials (inspired by Wigman's proof in the particular case of Legendre polynomials) and make extensive use of the recent literature on so-called fourth-moment theorems by Nourdin and Peccati. This is a joint work with Domenico Marinucci.

• Sartori Elena - Università di Padova

Strategic interaction in interacting particle systems. Slides

Models inspired by statistical mechanics have been vastly used in the context of social sciences to model the behavior of interacting economic agents. In particular, parallel updating models such as Probabilistic Cellular Automata have been proved to be very useful to represent rational agents aiming at maximize their utility in the presence of social externalities. What PCA do not account for is strategic interaction, i.e., the fact that, when deciding, agents forecast the action of other agents. In this talk I compare models that differ in the presence of strategic interaction and memory of past actions, exhibiting different stationary scenarios. The talk is based on joint work with Paolo Dai Pra and Marco Tolotti.

• Sobolieva Daryna - Taras Shevchenko National University of Kyiv

Large deviation principle for SDE's with discontinuous coefficients. Slides

In talk we will discuss different methods to prove large deviation principle for one-dimensional SDE's with discontinuous coefficients. It will be shown that discontinuity of coefficients leads, in general, to LDP asymptotics with rate function which differs from the rate function in the standard Freidlin-Wentzell result.

• Tantsiura Maksym - National Academy of Sciences of Ukraine

Motion with interaction of a particle system with infinite total mass. Slides

A stochastic differential equation describing the motion of a countable system of interacting particles is considered: $$\label{1}\left\{ \begin{array}{rll} dX_{k}(t)=& a(X_k(t),\mu_t)dt+dw_k(t),\ & k\in \mathbb{Z},\, t\in [0,T],\\ X_k(0)=& u_k,\ & k\in \mathbb{Z}\\ \mu_t=&\sum_{k\in \mathbb{Z}}\delta_{X_k(t)},&\\ \end{array} \right.$$ where $$a$$ is a bounded measurable function with a finite radius of interaction. A theorem of existence and uniqueness of the strong solution is proved. If the average concentration of $$\{u_k, k\in \mathbb{Z}\}$$ is not too high, assumptions of the theorem are satisfied. In particular, there exists a unique strong solution of the equation (\ref{1}) if $$\{u_k, k\in \mathbb{Z}\}$$ are atoms of a Poisson random measure with a uniform intensity.

• Zocca Alessandro - TU Eindhoven

Metropolis Markov chains for wireless random-access networks. Slides

We consider wireless random-access networks, modelled as systems of particles with hard-core interaction. Due to wireless interference, active users prevent other nearby users from simultaneous activity, which we describe as hard-core interaction on a conflict graph. The networks dynamics can be described as a Metropolis Markov chain and we examine the impact of torpid mixing and meta-stability issues on the delay performance of networks. The main focus is on finite grid networks, but our proof framework can be applied to other topologies as well, and is also relevant for the hard-core model in statistical physics and sampling from independent sets using single-site update Markov chains.