Curs 2016-2017: Abstracts i Slides

SEMINARS

21/09/16:  Constantinos Kardaras, London School of Economics, United Kingdom.

Viability and hedging in continuous-path markets with infinite number of assets.

We consider a continuous-time market with potentially infinite number of liquid continuous-path assets. Such abstraction is needed, for example, in the study of bond markets (where there is a continuum of maturities), markets with traded options (where there is a continuum of maturities and/or strikes), or in the post-limit study of large financial markets. In this general set-up, we provide exact versions of the fundamental theorem of asset pricing and hedging duality, completely analogous with the finite-asset case. Important in the development is a study of infinite-dimensional integration theory for continuous semimartingales which uses elements of Reproducing Kernel Hilbert Space theory.

05/10/16:  Giulia Binotto, Universitat de Barcelona, España.

Weak symmetric integrals with respect to the fractional Brownian motion.

We establish the weak convergence, in the topology of the Skorohod space, of the $\nu$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(\nu)\geq 1$ is the largest natural number satisfying $\int_0^1 \alpha^{2j}\nu(d\alpha)=(2j+1)^{-1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.

19/10/16:  Yuri Kabanov, Université Franche-Comté, Besançon, France.

Clearing in financial networks.

In the influential paper published in 2001, Eisenberg and Noe suggested a clearing procedure in the model describing a financial system composed of N banks. The asset of the bank consists in cash and exposures which are, in turn, liabilities for its debtors. The clearing consists in simultaneous paying all debts. Each bank pays to its counterparties the debts pro rata of their relative volume using its cash reserve and money collected from the credited banks. The rule is: either all debts are payed in full or the zero level of equity is attained and the bank defaults. The totals reimbursed by banks form an N-dimensional clearing vector. A remarkable feature is that this vector is a fixed point of a monotone mapping of a complete lattice into itself and its existence follows immediately from the Knaster--Tarski theorem, a beautiful and fairy simple result which proof needs only a few lines of arguments. The uniqueness of the clearing vector is a more delicate result involving the graph structure of the system. In the talk we discuss various generaizations: models with crossholdings, seniority of debts, default costs, illiquid assets etc.

26/10/16:  Mathew Joseph, University of Sheffield, United kingdom.

A discrete approximation to the stochastic heat equation.

We give a discrete space - discrete time approximation of the stochastic heat equation by replacing the Laplacian by the generator of a discrete time random walk and approximating white noise by a collection of i.i.d. mean 0 random variables. We give a few applications of this approximation, including fluctuations around the characteristic line for the harness process and the random average process.

16/11/16:  Annie Millet, Université Paris I, France.

On the Richardson acceleration of finite elements schemes for parabolic SPDEs.

We consider some finite elements approximations $u^h$ of the solution $u$ to a semi-linear parabolic SPDE whose coefficients satisfy the classical stochastic parabolicity condition, and $h>0$ is a scaling factor. Under proper conditions on the finite elements, we obtain an approximation of $u^h-u$ which is polynomial in $h$. This yields the corresponding Richardson acceleration method for such approximations of $u$ by $u^h$ given any prescribed speed. Some examples of finite elements which satisfy the assumptions are given. This is a joint work with I. Gyöngy (University of Edinburgh).

23/11/16:  Michel Ledoux, Université Toulouse-III-Paul-Sabatier, Toulouse, France.

How does the heat equation explore geometric and functional inequalities?

As is classical, the heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. As the prototypical parabolic partial differential equation and via its connection with Brownian motion in probability theory, the heat equation is of fundamental importance in diverse scientific fields, including (Riemannian) geometry, Lie groups, mathematical physics, graph theory, up to mechanics or biology. During the last decades, the gift of ubiquity of the heat equation has developed to approach various geometric and functional inequalities. Starting from the elementary example of Hölder’s inequality, the talk will feature illustrations of the heat flow method to families of integral inequalities, Sobolev inequalities and more refined isoperimetric theorems in Euclidean and Riemannian spaces. The method developed recently in discrete Boolean analysis motivated by problems in theoretical computer science.

14/12/16:  Dominique Bakry, Université Paul Sabatier, Toulouse III, France.

Representation of Markov kernels.

Consider some probability space $(E,\mu)$, and a given basis $(f_0=1,f_1,\dots,f_n,\dots)$ of $L^2(\mu)$. In many areas, ranging from PDE's to statistics, through computer algorithms and statistical physics, it is a major challenge to describe all sequences $(\lambda_n)$ such that the operator defined by $K(f_n)=\lambda_n f_n$ is a Markov operator, that is satisfies $K(1)=1$ and is positivity preserving. These sequences are called Markov sequences. This problem is completely solved for classical families of orthogonal polynomials (Hermite, Laguerre, Jacobi, for example). This is achieved through the description of extremal Markov sequences, which in many cases are just the sequences $\lambda_n=\frac{f_n(x)}{f_n(x_0)}$, for some $x_0\in E$. This property is called the hypergroup property at the point $x_0$. However, it is quite difficult in most situations to assert this property for a given model. In this talk, I shall describe a very powerful scheme to get this property, introduced in a paper of Carlen, Geronimo and Loss for the Jacobi polynomials. I shall show how this property applies to this case, and then extend this property to Dirichlet laws on the simplex.

25/01/17:  Lorenzo Zambotti, Université Pierre et Marie Curie, Paris, France.

The generalised KPZ equation.

In this talk we discuss a class of stochastic PDEs of parabolic type with one-dimensional space variable which has the nice property of being formally invariant under composition with a smooth function. One motivation is to construct a random string on a smooth finite dimensional manifold. At the moment the only way to do this is by regularity structures: in this context the renormalisation procedure can be nicely interpreted at the geometric level, since the renormalised quantities turn out to be invariant under change of local coordinates on the manifold. This is a joint work with Yvain Bruned and Martin Hairer.

25/01/17:  Raluca Balan, University of Ottawa, Canada.

Parabolic Anderson Model with space-time homogeneous Gaussian noise and rough initial condition.

The goal of this talk is to illustrate how various quantitative properties of the noise combined with the roughness of the initial condition may affect the existence of a random field solution of an SPDE, and to describe the impact of the noise and the initial condition on the behaviour of this solution. More precisely, we will consider the Parabolic Anderson Model on $\mathbb{R}_{+} \times \mathbb{R}^d$ driven by a space-time homogeneous Gaussian noise, with initial condition given by a signed measure. We assume that the covariance kernels of the noise in space and time are given by locally integrable non-negative-definite functions. We show that the solution to this equation exists and has a H\"older continuous modification, under the same respective conditions as in the case of the white noise in time. This shows that the temporal structure of the noise has no effect on the existence and H\"older regularity of the solution. However, the smoothness of the noise in time plays a big role in the order of magnitude of the moments of the solution.\\ This talk is based on joint work with Le Chen (University of Kansas).

25/01/17:  Robert Dalang, École polytechnique fédérale de Lausanne, Switzerland.

Global solutions to reaction-diffusion equations with super-linear drift and multiplicative noise.

We consider a stochastic heat equation with additive non-linearity $b$ and multiplicative non-linearity $\sigma$, in the case where the drift $b$ is super-linear: $\vert b(z)\vert \ge|z|(\log|z|)^{1+\varepsilon}$, for some $\varepsilon>0$. When $\sigma\equiv 0$, it is well known that such PDEs frequently have non-trivial stationary solutions. By contrast, Bonder and Groisman (2009) have shown that when $\sigma$ is constant and $\sigma \neq 0$, there is finite-time blowup. We prove that the Bonder-Groisman condition is unimproveable by showing that the reaction-diffusion equation with noise is ''typically'' well posed when $\vert b(z) \vert =O(|z|\log_+|z|)$ as $|z|\to\infty$. We interpret the word ''typically'' in two essentially-different ways without altering the conclusions of our assertions. This is a joint work with Davar Khoshnevisan (University of Utah) and Tusheng Zhang (University of Manchester).

25/01/17:  Ciprian Tudor, Université de Lille I, France.

Connection between the number theory and Malliavin calculus.

The study of the zeros of the Riemann zeta function constitutes one of the most challenging problems in mathematics. A large literature is devoted to the study of the behavior of the zeta zeros. We will discuss how tools from stochastic analysis, and in particular from Stein-Malliavin calculus, can be used in the study of the zeta function.

01/03/17:  Maria Jolis, Universitat Autònoma de Barcelona, Catalonia.

Intermittency properties of the solution of the stochastic wave equation with linear multiplicative fractional noise with index $H<1/2$.

In two recent articles we studied the stochastic wave and heat equation with affine multiplicative Gaussian noise that is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index $H\in (1/4,1/2)$. We proved the existence and uniqueness of solutions and their Hölder continuity properties. In this seminar, we consider the stochastic wave equation on the real line driven by a linear multiplicative noise (the so called hyperbolic Anderson model) with constant initial data. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the $p$-th moment of the solution, for any $p\geq 2$. We also see that condition $H>1/4$ turns out to be necessary for the existence of solution. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent. The talk is based on joint work with Raluca Balan (Univ. Ottawa) and Lluís Quer (UAB).

15/03/17:  Marta Sanz-Solé, Universitat de Barcelona, Catalonia.

Systems of stochastic Poisson equations.

We will introduce a system of elliptic SPDEs with additive white noise. Properties of the sample paths of the solution and of the probability law will be discussed. Partial results on hitting probabilities will be presented.

22/03/17:  André Süss, Credit Suisse, Zürich, Switzerland.

A Multi Interest Rate Curve Model for Exposure Modelling.

The tenor basis phenomenon became significant with the 2007 financial crisis and has altered the traditional way of one-curve pricing and risk management to a multi-curve phenomenon. The stochastic nature of basis spreads between curves particularly poses a challenge for forward looking applications like XVA or real world measure exposure analytics. In this talk, we discuss a two-factor Gaussian approach for modelling multiple fixing curves and basis spreads in the risk neutral and spot measure, show the impact on basis swap exposure, investigate the correlation structure and discuss the pros and cons of interpreting as a spread or multi curve model respectively. This is based on a joint work with A. Boldin, R. Lichters and M. Trahe which is available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2870698 .

05/04/17:  María José Garrido, Universidad de Sevilla, España.

Exponential Stability of solutions to SPDEs driven by fractional Brownian motion.

In this talk we are concerned with the study of the longtime behaviour of stochastic evolution equations driven by a fractional Brownian motion (fBm). We will analyze different approaches and consider both the cases of an fBm with Hurst parameter $H\in (1/2,1)$ and $H\in (1/3,1/2]$. This talk is based on some joint works with A. Neuenkirch (University of Mannheim, Germany) and B. Schmalfuss (University of Jena, Germany).

03/05/17 - 31/05/2017:  Rosario Delgado, Universitat Autònoma de Barcelona, Catalonia.

Xarxes Bayesianes: una eina probabilística per al perfilat i la classificació.

Les Xarxes Bayesianes són models probabilístics molt utilitzats per a sistemes complexos en els que intervenen un gran nombre de variables, ja que permeten modelitzar les relacions de dependència entre elles. S'han fet servir amb èxit en molts àmbits, com economia, medi ambient, enginyeria, valoració de riscos, biologia, criminologia, etc. En aquesta xerrada s'explicarà què són aquests models, quins mètodes es fan servir per a l'estimació dels seus paràmetres i com es fa inferència a partir d'ells. A partir d'una aplicació concreta real, es presentarà el seu ús per al perfilat delinqüencial i per a fer classificacions.

17/05/17:  Michael Coulon, University of Sussex, United Kingdom.

Environmental Market Design: Understanding Certificate Price Dynamics and the Impact of Regulatory Policy.

Environmental markets for tradable certificates come in several categories, including both carbon emissions allowances and green (renewable energy) certificates. In both cases, a variety of market designs and structures exist, with rules sometimes even changing frequently within a given market. Relatedly, several fundamental weaknesses of environmental markets have been witnessed through history, including sudden price swings and extended periods of over- or undersupply, eroding market confidence. Such price behavior and regulatory uncertainty illustrates the need to investigate alternative design ideas and their likely impact on market dynamics. In this talk, we introduce several policy proposals which have been adapted to and tested on the New Jersey solar renewable energy certificate (SREC) market, together with a stochastic structural model for prices and solar generation. In particular we discuss the use of a sloped compliance penalty function instead of a traditional 'cliff' or binary function, together with a dynamic and adaptive mechanism for setting future requirement levels as a function of previous surpluses or shortages. We analyze the optimal decisions of market participants facing SREC submission and banking decisions, and compare the resulting implications for prices. Finally, we discuss briefly some related recent work on the design of a new global carbon market, which also embeds the crucial feature of responding to unexpected shocks via adaptive emissions reduction targets, this time with an additional focus on an equitable and transparent distribution of responsibilities across countries.

24/05/2017:  Liliana Peralta Hernández, Universidad Autónoma del Estado de Hidalgo, México.

Un criterio de Osgood para una ecuación diferencial estocástica semilineal.

Cuando los coeficientes de una ecuación diferencial estocástica no satisfacen la restricción de crecimiento lineal ésta puede tener una solución que explota en tiempo finito, es decir, las trayectorias de la solución pueden “tender” a $\pm\infty$ cuando el parametro del tiempo $t$, se aproxima a un tiempo aleatorio finito. Para el caso de ecuaciones diferenciales estocásticas y deterministas cuyos coeficientes no dependen del tiempo existen criterios como la prueba de Feller y la de Osgood, respectivamente, para determinar si un proceso explota o no en tiempo finito. La propuesta de esta platica es presentar extensiones de la prueba de Feller y el criterio de Osgood para la explosión en tiempo finito de la ecuación diferencial estocástica semilineal con coeficientes dependientes del tiempo de la forma $X_t=\xi+\int_0^t b(s,X_s)\,ds+\int_0^t \sigma(s)X_s\,dW_s\,ds \quad t\geq0,$ donde, $b$ es no negativa y no decreciente por componentes, $\sigma$ es un proceso predecible y continuo, $W$ es un $\mathcal{F}_t$-movimiento Browniano y $\xi$ es una variable aleatoria $\mathcal{F}_0$-medible.

21/06/2017:  David Ruiz Baños, University of Oslo, Norway.

A type of fractional noise that regularizes flows of SDEs with discontinuous coefficients.

28/06/2017:  David Nualart, University of Kansas, USA.

Noncentral limit theorem for the generalized Rosenblatt process.

In this talk we present a noncentral limit theorem for the divergence operator in the framework of Malliavin calculus. As an application we derive the convergence in law of a family of generalized Rosenblatt processes with kernels defined by a parameter taking values in a tetrahedral region of the q-dimensional Euclidean space, when the parameter converges to a face of the region. The convergence in stable and the limit is a compound Gaussian distribution with random variance given by the square of a Rosenblatt process of one lower rank.

28/06/2017:  Archil Gulisashvili, Ohio University, USA.

Gaussian Selfsimilar Stochastic Volatility Models.

The results discussed in the talk are joint with F. Viens and X. Zhang. The talk is devoted to uncorrelated Gaussian selfsimilar stochastic volatility models. The volatility of an asset in such a model is described by the absolute value of a selfsimilar Gaussian process. A typical example of such a volatility process is fractional Brownian motion. We obtain sharp asymptotic formulas describing the small-time behavior of the asset price density, the call pricing function, and the implied volatility in a Gaussian selfsimilar model. The parameters appearing in the asymptotic formulas mentioned above are expressed in terms of the Karhunen-Loeve characteristics of the volatility process. We will discuss examples of Gaussian selfsimilar stochastic volatility models, and show how to recover the selfsimilarity index knowing the small-time behavior of the call pricing function or the implied volatility.