Curs 2015-2016: Abstracts i Slides


06/07/16:  Jan Pospisil, University of West Bohemia, Pilsen, Czech Republic.

Lessons learned from stochastic volatility models calibration and simulation.

Stochastic volatility (SV) models are used in mathematical finance to evaluate derivative securities, especially options. The aim of this talk is to present our experiences and best practices with calibration of continous-time SV models to real-market data and with Monte Carlo simulations of these models. In particular, popular Heston and Bates models and newly proposed approximative fractional SV models will be discussed.

27/04/16:  Jan Pospisil, University of West Bohemia, Pilsen, Czech Republic.

Unifying approach to several stochastic volatility models with jumps.

In this talk we introduce a new unifying approach to option pricing under continuous-time stochastic volatility models with jumps. For European style options, a semi-closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro-differential equation (PIDE). There exist transforms for a wide set of possible option payoffs and once the so called fundamental transform of a given model is obtained it can be used to price different European options. By finding the fundamental transform of a general jump diffusion model we introduce a formula for a wide class of stochastic volatility models where several different kinds of jumps can be involved. We discuss the numerical performance of the proposed formula.

20/04/16:  Alexandre Richard, INRIA Sophia-Antipolis, France.

Continuity in the Hurst parameter of certain functionals of fractional Brownian motion.

The fractional Brownian motion is now widely used in many models, and although the statistical determination of H is well-studied problem, the sensitivity in the Hurst parameter of the model is in general unknown. In this talk, we will present several examples where the Hölder continuity in the Hurst parameter can be established. Our examples include a family of fractional Brownian fields and the hitting times of fractional diffusions.

06/04/16:  Salvador Ortiz, University of Oslo, Norway.

A second order time discretization of the nonlinear stochastic filtering problem.

Partially observed dynamical systems are ubiquitous in a multitude of real-life phenomena. The dynamical system is typically modelled by a continuous time stochastic process called the signal process. The signal process cannot be measured directly, but only via a related process, called the observation process. The filtering problem is that of estimating the current state of the dynamical system at the current time given the observation data accumulated up to that time. The solution of the continuous time filtering problem can be represented as a ration of two expectations of certain functionals of the signal process that are parametrized by the observation path. In this talk we introduce a new discretization of these functionals, corresponding to a chosen partition of the time interval, and show that the convergence rate of the discretization is proportional to the square of the mesh of the partition. We hope to use this discretization, in conjunction with high order approximations of the law of the signal, to design high order particle filters. This is a joint work with Dan Crisan (Imperial College London).

02/03/16:  David Ruiz Baños, IMUB.

Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift.

In this talk we present a new method for the construction of strong solutions of SDE’s with discontinuous drift coefficients driven by a multidimensional fractional Brownian motion for small Hurst parameters. Furthermore, we prove the rather surprising phenomenon of the higher order Fréchet differentiability of stochastic flows of such SDE’s in the case of a small Hurst parameters. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can also be applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths. This is a joint work with Prof. Frank Proske and Dr. Torstein Nilssen (University of Oslo).

20/01/16:  Giovanni Peccati, University of Luxembourg.

Non-Universal second order behaviour of arithmetic random waves.

Originally introduced by Rudnick and Wigman, arithmetic random waves are Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk, I will describe the asymptotic behaviour of the so-called "nodal length" (that, is the volume of the zero set) of such random objects, and show that (quite unexpectedly) it is non-central and non-universal. Joint work with D. Marinucci (Rome Tor Vergata), M. Rossi (Luxembourg) and I. Wigman (King's College, London).

09/12/15:  Francisco Delgado.

Spectral-based numerical method for Kolmogorov equations in Hilbert spaces.

We propose a numerical solution for the solution of the Fokker-Planck-Kolmogorov (FPK) equations associated to stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein-Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as a infinite system of ordinary differential equations and by truncation, we set a linear finite system of differential equations. The solution of such system allows us to build an approximation to the solution of the Kolmogorov equations. We will apply the numerical method to the Kolmogorov equations associated to a stochastic diffusion equation, a Fisher-KPP stochastic equation and the stochastic Burgers Eq. in dimension 1.

18/11/15:  Mohammud Foondun, Loughborough University, UK.

Approximations of a class of stochastic heat equation.

In this talk, we will look at how one can approximate a class of stochastic heat equation by infinite dimensional interacting SDEs. We will also discuss how one can then transfer results about these SDEs to the SPDE. For example, we will show how to obtain moment comparison principle. This is joint work with Mathew Joseph and Shiu-Tang Li.

18/11/15:  Ciprian Tudor, Université de Lille, France.

On the determinant of the Malliavin matrix on Wiener chaos .

A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order, we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional and that the result is not true for random variables in Wiener chaoses of different orders.

28/10/15:  Anthony Reveillac, INSA, Toulouse, Francia.

Stochastic regularization effects of semi-martingales on random functions.

In this talk we address an open question formulated by Flandoli, Gubinelli and Priola concerning the extention of the so-called Itô-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and the solution of the Fokker-Planck equation associated to X, to random mappings f. To this end we provide new results on a class of adapted and non-adapted Fokker-Planck SPDEs and BSPDEs and we make use of the Malliavin calculus. This is a joint work with Romain Duboscq.

14/10/15:  Allan Fiel Espinosa, CINVESTAV-IPN, México D.F.

Stability for a class of stochastic fractional systems.

We obtain a closed expression for the solution of a linear Volterra integral equation with an additive Hölder continuous noise, which is a fractional Young integral, and with a function as initial condition. This solution is given in terms of the Mittag-Leffler function. Then we study the stability of the solution via the fractional calculus. As an application we analyze the stability in the mean of some stochastic fractional integral equations with a functional of the fractional Brownian motion as an additive noise. Also, we extend these results to the case that the involved equation is semilinear.

07/10/15:  David Ruiz Baños, University of Oslo, Oslo, Noruega.

Optimal bounds and Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients.

We consider a process given as the solution of a one-dimensional stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Hölder continuity of any order of the Lebesgue density of that process at any given time is achieved. Explicit and optimal upper and lower bounds for the densities are also obtained. The regularity of the densities is obtained via the inverse Fourier theorem by identifying the stochastic differential equation with the "worst" characteristic function. The latter is done by employing a method already introduced by the authors in a previous work to find explicit optimal bounds for the densities. Then we generalize our findings to a larger class of diffusion coefficients. In the Markovian setting the densities are directly connected to the fundamental solution of the Fokker-Planck equation. For this reason, this entitles us to improve the regularity of the fundamental solution in dimension one. This is a joint work with Dr. Paul Krühner.

30/09/15:  Jorge A. León, CINVESTAV-IPN, México D.F.

Extension of the Young's integral via fractional calculus.

In this talk we study the Young's integral of an Hölder continuous function with exponent k with respect to fractional Brownian motion with parameter H such that H(k+1)<1.