Curs 2011-2012: Abstracts i Slides

26/06/12: Albert Ferreiro-Castilla, University of Bath, Anglaterra.

Multilevel Monte Carlo simulation for Lévy processes based on the Wiener-Hopf factorization.

We will present the Multilevel Monte Carlo (MC) method introduced by Giles (2008) and combine this technique with the MC simulation technique for Lévy processes introduced in Kuznetsov et al. (2011). The results reveal that the coupled method attains optimal convergence rates. One of the key properties of the method is the independence with respect to the nature of the jump component as opposed to standard approaches to simulate Lévy processes by a compound Poisson process, whose rate of convergence typically depends on the Blumenthal-Getoor index.

References:

[1] Giles, M.B. (2008) Multilevel Monte Carlo Path Simulation.

[2] Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K. (2011) A Wiener-Hopf Monte Carlo simulation technique for Lévy process.

[Slides]

30/05/12: Frederi Viens, Purdue University, USA.

New Malliavin Calculus techniques for non-Gaussian comparisons.

The stochastic calculus of variations of Paul Malliavin has found applications in a surprisingly wide array of topics in mathematics, statistics, and other fields. We discuss a recent development in the application of this Malliavin calculus to provide quantitative estimates for random variables on Wiener space. We will see how the notion of Gaussian distribution and of covariance can be generalized by using Malliavin calculus operators, which are then used to extend classical theorems to non-Gaussian fields, including the Fernique-Sudakov comparison for expected suprema, and the Gordon-Slepian comparison of appropriately convex Gaussian functionals. Applications to the stochastic heat equation and the Sherrington-Kirkpatrick model will be given.

This is joint work in progress with Ivan Nourdin and Giovanni Peccati, and is based on recent advances by these two authors, and work by the presenter, in how to use the Malliavin calculus for Gaussian comparisons.

[Slides]

09/05/12: David Applebaum, University of Sheffield, Anglaterra.

Second quantised representation of Mehler semigroups associated with Banach space valued Levy processes.

The solutions of linear SPDEs driven by Banach space valued additive Levy noise are generalised Ornstein-Uhlenbeck processes. These are Markov processes and their transition semigroups are sometimes called Mehler semigroups. If the driving noise is a Brownian motion then Anna Chojnowska-Michalik and Ben Goldys have shown that these semigroups can be represented by means of second quantisation within a suitable chaotic decomposition. The result has recently been extended to the Levy case (for Hilbert space valued noise) by Szymon Peszat using a point process construction. In this talk I will present an alternative approach to this construction based on the use of exponential martingales.

26/04/12: Archil Gulisashvili, Ohio University, USA.

Asymptotic behavior of implied volatility at extreme strikes in stochastic stock price models.

The talk concerns the asymptotic behavior of the Black-Scholes implied volatility as a function of the strike price. We consider a general stochastic stock price model under a risk-neutral measure, and establish sharp asymptotic formulas with error estimates for the implied volatility at large and small strikes for such a model. These formulas are rather universal. On the one hand, they imply well-known results, e.g., R. Lee's moment formulas and the tail-wing formulas due to S. Benaim and P. Friz. On the other hand, sharp asymptotic formulas allow us to characterize the asymptotic behavior of the implied volatility in various special models. It is worth mentioning that the behavior of the implied volatility is qualitatively different in models with moment explosions and in models, for which all the moments of asset price are finite. It will be explained in the talk how the implied volatility behaves in several classical stochastic volatility models (Hull-White, Stein-Stein, Heston). These models belong to the group of models with moment explosions. For stochastic stock price models without moment explosions, V. V. Piterbarg conjectured an asymptotic formula, which can replace R. Lee's moment formula at large strikes. We will show that Piterbarg's formula is valid in a slightly modified form, and also confirm Piterbarg's original conjecture under very mild restrictions. Finally, we will characterize the asymptotic behavior of the implied volatility in special stock price models without moment explosions.

25/04/12: Archil Gulisashvili, Ohio University, USA.

Asymptotic behavior of stock price distribution densities in stochastic volatility models.

Stock price models with stochastic volatility have been developed in the last decades to improve pricing and hedging performance of the classical Black-Scholes model and to account for certain imperfections in it.

The following stochastic volatility models will be considered in the talk: the Hull-White model, the Stein-Stein model, and the Heston model. We will establish sharp asymptotic formulas with relative error estimates for the stock price densities in these models. The formulas show that the distributions of the stock price in the models mentioned above have Pareto type tails, that is to say, the tails decay like regularly varying functions. Once we have a good understanding of how the stock price density changes, many other characteristics of stochastic volatility models can be analyzed. We will illustrate the previous statement by deriving sharp asymptotic formulas for call option pricing functions and for implied volatility at extreme strikes in the Hull-White, Stein-Stein, and Heston models.

28/03/12: Marco Ferrante, Università di Padova, Itàlia.

Convergence results for Particle Filtering approximations and their extensions to the case of unbounded test functions.

The stochastic filtering problem deals with the estimation of the current state of a signal process given the observation supplied by an associate process, called the observation process. Particle filtering method are a set of flexible and powerful sequential Monte Carlo methods designed to solve the filtering problem numerically. I my talk I will present some known results about the convergence of this approximate solution to the true one and some new extensions to the case of unbounded test functions, which allow to prove convergence of the moment of the approximate distribution to the moments of the true one.

[Slides]

22/02/12: Giulia di Nunno, University of Oslo, Noruega.

Dynamic no-good-deal pricing measures.

The fundamental theorem of asset pricing is the key result celebrating the marriage between the economic principle of no-arbitrage and the mathematical tools of martingales and equivalent martingale measures. These provide the fundamental framework for pricing. Several versions of this outstanding result have appeared with progressive improved level of generality.

A crucial observation is that, provided existence, there is no uniqueness of equivalent martingale measure guaranteed with the exception of markets that are complete, namely, in markets where all claims are attainable. However, it is well-known that such markets are more a mathematical abstraction than proved reality and in general markets have to be considered incomplete. As a consequence the problem of selecting one equivalent martingale measure out of the infinite many available has been largely treated. The literature in this direction is vast.

More recently a new idea was developed: instead of selecting a single measure, one can restrict the set of equivalent martingale measures characterizing those that are in some sense reasonable". The criteria suggested so far are three:

- to restrict to equivalent martingale measures that rule out deals that are "too good to be true"

- to restric the set of equivalent martingale measures by choosing those with a density lying within pre-considered lower and upper bounds

- to restrict the set of equivalent martingale measures to those compatible with bid and ask bounds observed for some traded options

In this talk I will discuss the first criteria and I will focus on linear price systems in incomplete markets that are consistent with lower and upper bounds set on the Sharpe ratio. The results show the existence of an equivalent martingale measure that both represents the prices and guarantees the non availability of deals that are too good to be true.

However, our methods can serve the other criteria as well. In particular our results include extension theorems for operators (stochastic processes) that are sandwich preserving, and representation theorems in terms of conditional expectations. The relationship with the existence of probability measures and its links with the extensions. The methodologies presented rely on probability and functional analysis, and are also mixed with techniques in fashion in the theory of risk measures.

The talk is based on the paper: Jocelyne Bion-Nadal and Giulia Di Nunno (2011). Extension theorems for linear operators on $L_\infty$ and application to price systems. ArXiv: 1102.5501v1.

[Slides]

25/01/12: Maria Emilia Caballero, Instituto de Matematicas UNAM, Ciudad de México.

Procesos de Lévy-estables e hipergeométricos.

Se presentaran varios ejemplos de procesos de Lévy que se contruyen a partir de procesos de Markov autosimilares positivos usando la representación de Lamperti. Se podran estudiar, en algunos casos, la descomposicion de Wiener-Hopf y en otros dar propiedades tipo leyes de salida y otras características de los procesos construidos.

[Slides]

21/12/11: Eulàlia Nualart, Université Paris XIII, Paris.

Estimació de densitats de processos de difusió amb salts.

[Slides]

15/11/11: Rahul Roy, Indian Statistical Institute, New Delhi.

Finite clusters in a high intensity Zwanzig percolation model.

Zwanzig (1963) studied a system of non-overlapping hard needles in the continuum, where the orientation of the needles were restricted to a finite set. Here he observed that as the density of needles increased a phase transition occurred from an isotropic phase, where the rods are placed `chaotically', to a nematic phase, where the rods are oriented in a fixed direction. We formalise this model through a study of the structure of finite connected components in a high density supercritical regime and thereby establish the nematic behaviour as observed by Zwanzig.

[Slides]

09/11/11: Rahul Roy, Indian Statistical Institute, New Delhi.

Random directed spanning trees and the Brownian web.

Random directed spanning trees are random graphs with vertices being integer lattice points chosen according to an i.i.d. Bernoulli distribution and edges between vertices being along a particular direction and also according to a prescribed minimality condition. Drainage networks are modeled by physicists as such random directed trees. We study such trees in various dimensions. We also study the scaling limit of such trees in 2-dimensions and show that they agree with the scaling limit of coalescing random walks, i.e. the Brownian web.

[Slides]

26/10/11: Ahmadou Bamba Sow, Gaston Berger University, Senegal.

Homogenization of PDE with periodic coefficients: A probabilistic approach.

In order to study the physical phenomena occurring in nature, we first need a model to determine the equations governing these phenomena, and then we need to calculate (numerically) the solutions of these equations. But sometimes the computing power that is available is not sufficient, not because of some algorithms developed but due to the inherent complexity of the problem, for example when the environment in which the physical phenomenon is studied present some complexities.

The aim of the theory of homogenization is to approximate homogeneous (in a simple way) a medium described by the microscopic properties assumed heterogeneous. Fields of application are varied: the study of basement (diffusion of oil in porous media), properties of composite materials, ceramic materials, superconducting materials suprafilamentaires, study of polymers ... For example, the study of heat diffusion in these environments there will be modeled as follows:

$\partial_t u^\epsilon (t, x) = \frac{1}{2} \mathrm{div}_x \big[a(x/\epsilon) \nabla u^\epsilon (t, x)\big ]$

with a given initial condition of the type

$u^\epsilon(0, x) = f (x).$

The parameter $\epsilon$ is introduced to account for the small size of the heterogeneities of the medium. To find an approximation of the solution u, the objective is to tend $\epsilon$ to 0 and identify a possible limit. The formula of Kac-Feynman gives the relationship between the analytical point of view and the probabilistic approach in this case. In this talk I will present a brief review of known results and then present two recent studies with weaker assumptions on the coefficients.

[Slides]

28/09/11: Jorge León, CINVESTAV, Mèxic.

Un criterio de Osgood para ecuaciones integrales perturbadas por un ruido aditivo.

En esta charla usaremos un criterio de comparación para mostrar que el criterio clásico de explosión de Osgood sigue siendo válido para cierto tipo de ecuaciones integrales perturbadas por un ruido aditivo. Como aplicación veremos que la ley del logaritmo iterado implica que las trayectorias del bifraccionario son ejemplos de ruido que cumplen las condiones pedidas para nuestro criterio de explosion en tiempo finito de las soluciones".