Curs 2012-2013: Abstracts i Slides

26/06/13: Archil Gulisashvili, Ohio University, USA.

The Heston Riemannian distance function.

This is a joint work with P. Laurence. The talk concerns the Heston model that is one of the most popular stock price models with stochastic volatility. We study the Riemannian distance function associated with the log-price process and the variance process in the Heston model, using analytical and geometrical methods. Explicit formulas are provided for the Heston distance function. The analytical methods used in the present work are based on close relations between the Heston model and a simpler model that we call the Grushin model. In addition, a partial large deviation principle is established for the Grushin model.

[Slides]

19/06/13: Mark Veraar, Technical University Delft, NL.

A new approach to stochastic evolution equations with adapted drift.

The variation of constants/semigroup approach to stochastic evolution equations can easily be extended to the case the drift/generator A(t) is time dependent. Here the semigroup is usually replaced by a two parameter family which is called the evolution family in the literature. If A depends on time and the probability space in an adapted way this approach seems to fail dramatically: The integrands of the stochastic integrals become non-adapted and one cannot argue as in the Ito setting. In a paper by Leon and Nualart, the theory of Skorohod and forward integration is used to extend this approach with non-adapted integrands. Here the evolution family is assumed to have certain Malliavin differentiability properties. In the talk I will present a new representation formula for the solution in the parabolic setting. This formula allows to study the equations more directly and obtain stronger regularity results for the solution. This is based on joint work with Matthijs Pronk.

[Slides]

12/06/13: José Fajardo, Getulio Vargas Foundation (Río de Janeiro).

Barrier Options Under Lévy Processes: A Simple Short-Cut.

In this paper we present a simple way to price a class of barrier options when the underlying process is driven by a huge class of Lévy processes. To achieve our goal we assume that our market satisfies a symmetry property. In case of not satisfying that property some approximations and relationships can be obtained.

[Slides]

29/05/13: Jorge León, CINVESTAV, Mèxic.

Lyapunov stability of fractional systems.

In this talk we study the stability of autonomuous fractional systems using conditions of integer systems. Joint work with Rafael Martinez.

[Slides]

15/05/13:  Marco Ferrante, Università degli Studi di Padova, Itàlia.

Markovian sports: Tennis vs. Volleyball.

In this talk I will consider tennis and volleyball under the Markovian assumption that the probability of winning a single rally is independent of the other rallies and constant during the game. Fixing two parameters which indicate the probabilities of winning a rally for the serving player or team, I am able to derive the exact value of the probability of winning a set (and a match) and the mean duration of the games. While the results for the tennis are classic, except for the duration part, those for the volleyball, considered in in the current rally point- and in the former side-out scoring systems, are to the best of my knowledge new. In the case of volleyball, I am able to prove that the present point system reduces the winning probability of the stronger team, adding interest/randomness to the game, and that the mean duration of the games in the current scoring system, as well known in the practice, has reduced the (expected) length of the matches, making the game more spectator- and television-friendly.

The empirical cost of optimal incomplete transportation.

We consider the problem of optimal incomplete transportation between the empirical measure on an i.i.d. uniform sample on the d-dimensional unit cube, $[0,1]^d$ , and the true measure. This is a family of problems lying in between classical optimal transportation and nearest neighbor problems. We show that the empirical cost of optimal incomplete transportation vanishes at rate $O_P(n^{-1/d})$, where n denotes the sample size. In dimension $d\geq3$ the rate is the same as in classical optimal transportation, but in low dimension it is (much) higher than the classical rate.

[Slides]

13/02/13 - 13/05/13:  Jan van Neerven, Technical University Delft, NL.

Stochastic integration in UMD Banach spaces.

10 sessions of 1h30min. For the contents see the description.

19/12/12:  David Nualart, University of Kansas, Estats Units.

Symmetric fractional Brownian motion.

In this talk we present some results on the properties of the eigenvalues of a symmetric random matrix whose entries are fractional Brownian motions with Hurst parameter H>1/2.

[Slides]

12/12/12: Arnaud Debussche, ENS Cachan Bretagne, França.

Existence of densities for non smooth SDEs.

We propose a new method to prove existence of densities for the law of solutions of SDEs with non smooth coefficient. It is an extension of a method proposed by Fournier and Printems which could be applied only for one dimensional processes. Using Besov spaces, we are able to consider SDEs driven by stable like Levy noises and finite dimensional projection of solutions of the Navier-Stokes equations in dimension 3. We prove existence of densities with Besov regularity.

28/11/12:  Eulàlia Nualart, Universitat Pompeu Fabre, Espanya.

Gaussian estimates for the density of the Landau SDE.

The Landau SDE is a nonlinear stochastic differential equation driven by a space-time white noise, introduced by Guérin in order to give a probabilistic interpretation of the spatially homogeneous Landau equation, which is a nonlinear partial differential equation that describes the movement of particles in a plasma, widely studied by Desvillettes and Villani, among others. Guérin showed existence and uniqueness of the solution to the Landau SDE, as well as existence and smoothness of the density by means of the Malliavin calculus. In this talk, we will show how to obtain Gaussian bounds for this density, which continues the work started by Guérin, Méléard and Nualart.

This is a joint work with François Delarue (Nice) and Stéphane Menozzi (Évry).

[Slides]

21/11/12:  Noèlia Viles, Universitat de Barcelona, Espanya.

Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals.

A continuous time random walk (CTRW) is a pure jump process given by a sum of i.i.d. random jumps separated by i.i.d. random waiting times (positive random variables). CTRW became a widely used tool for describing random process that appear in a large variety of physical models and in finance.

We prove a functional limit theorem for the quadratic variation of the CTRW under distributional assumptions.

In the talk, I will also discuss a recent progress in the convergence of the stochastic integrals of a deterministic function driven by a time-changed symmetric $\alpha$-stable Lévy process. The motivation of this problem comes from the physical model given by a damped harmonic oscillator subject to a random force studied in the paper of Sokolov [3].

This is a joint work with E. Scalas.

References:

[1] Scalas, E.; Viles, N: On the Convergence of Quadratic variation for Compound Fractional Poisson Processes. Fractional Calculus and Applied Analysis, 15, 314–331,(2012).

[2] Scalas, E.; Viles, N.: A functional limit theorem for stochastic integrals driven by a time-changed symmetric $\alpha$-stable Lévy process. Work in progress.

[3] Sokolov, I.M.: Harmonic oscillator under Lévy noise: Unexpected properties in the phase space. Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011).

31/10/12:  Charles Newman, Courant Institute, New York.

Optimization and Spin Glasses in Dimension d.

Take the graph whose vertices are the integer points in an N x N x...x N d-dimensional cube and whose edges are between pairs of nearest-neighbor vertices. Assign random values to the edges and among all subgraphs that touch each vertex of the cube and have no loops (i.e., spanning trees), let $T_N$ be the one for which the sum of the edge variables is minimum. The limit as $N\rightarrow\infty$ of this Minimal Spanning Tree (MST) is either a single spanning tree or a forest of many trees. Which is the case should depend on the dimension d, but the only rigorous complete results are that for d=1 and 2, it is a single tree. We will discuss the relation between results and conjectures for the MST and those for spin glasses, which are mathematical models that originated in condensed matter physics but are being used elsewhere such as sociological models.

[Slides]

24/10/12:  Arturo Valvidia, Universitat de Barcelona, Espanya.

Pricing Contingent Convertibles Bonds: a Credit Risk Approach.

We study the problem of pricing debt instruments known as Contingent Convertibles (CoCos). A CoCo is converted into the equity of the issuing bank upon the appearance of a trigger event. This trigger mechanism provides an automatic strengthening of the capital structure of the bank. Equity is in this case injected on the very moment the bank is failing to meet the minimum regulatory capital requirements or when it is heading towards a state of non-viability. The time at which the trigger occurs resembles a default time. Consequently, we can derive explicit CoCo's pricing formulae, from both the Structural and Reduced-Form approaches to Credit Risk modelling. In this talk present the main mathematical challenges and tools related to these computations.

[Slides]

03/10/12:  Andre Suess, Universitat de Barcelona, Espanya.

Integration theory for infinite dimensional volatility modulated Volterra processes.

In this talk we present a stochastic integration theory with respect to a possibly infinite-dimensional, volatility modulated Volterra process (VMV). This extends the results in Alòs, Mazet and Nualart (2001) to allow for stochastic volatility and Barndorff-Nielsen el al. (2012) to be infinite dimensional. The integration operator is based on Malliavin calculus and leads to an anticipative integral. In this talk we motivate the relevant integral, derive its fundamental properties, give some applications and show some open problems and further lines of research.

26/09/12:  Jorge León, CINVESTAV, Mèxic.

Ecuaciones diferenciales lineales anticipantes gobernadas por procesos de Lévy.

En esta charla usaremos las ideas desarrolladas por Buckdahn para estudiar la existencia de una única solución de ecuaciones estocásticas lineales gobernadas por un proceso de Lévy X. Aquí los coeficientes y la condición inicial no son necesariamente adaptados a la filtración dada, y las integrales con respecto a la parte continua y la parte discontinua de X son en el sentido de Skorohod y trayectoriales, respectivamente.

19/09/12:  Monique Jeanblanc, Evry Val d'Essonne University, France.

The Azema supermartingale.

In this paper, we prove that, if an $F$-supermartingale $Z$ valued in [0,1] admits a multiplicative decomposition $Z_{t}=N_{t}e^{-A_{t}}$ where $A$ is a continuous increasing process, one can construct (infinitely many) random times $\tau$ such that $P(\tau>t|F_{t})=N_{t}e^{-A_{t}}$. All these random times admit the same intensity process $\Lambda_{t}=A_{t}$.

Key Words: credit risk, enlargement of filtration, intensity, Predictable Representation Theorem, multiplicative decomposition of supermartingales

17/09/12:  Amel Bentata, Institut für Mathematik Universität Zürich, Suissa.

Markovian Projection of Stochastic Processes.

We give conditions under which the flow of marginal distributions of a discontinuous semimartingale can be matched by a Markov process whose infinitesimal generator can be expressed in terms of its local characteristics, generalizing a result of Gyongy (1986) to the discontinuous case. Our construction preserves the martingale property and allows to derive a partial integro-differential equation for the one-dimensional distribution of discontinuous semimartingales, extending the Kolmogorov forward equation (Fokker Planck equation) to a non-Markovian setting.

[Slides]