Curs 2018-2019: Abstracts i Slides



05/06/19: Marta Sanz Solé. UB-BGSMath

A stochastic wave equation with super-linear coefficients

We consider the stochastic wave equation on Rd, d ∈ {1, 2, 3},

2 /∂t2u(t,x)2 /∂x2 u(t,x)=b(u(t,x))+σ(u(t,x))W(t,x), t(0,T],

u(0, x) = u0(x), /∂t u(0, x) = v0(x), (1) 

where for d = 1, W ̇ is a space-time white noise, while for d = 2,3, W ̇ is a white noise in time and correlated in space. The coefficients of the equation, b and σ, satisfy

|σ(x)| ≤ σ1 + σ2|x| ln+(|x])a, |b(x)| ≤ θ1 + θ2|x| ln+(|x])δ, (2)

where θii R+, i = 1,2, σ2 ̸= 0, δ,a > 0, with b dominating over σ. We are interested in the well-posedness of (1), motivated by the recent work [R. Dalang, D. Khoshnevisan, T. Zhang, AoP, 2019] on a one-dimensional reaction-diffusion equation with super-linear coefficients satisfying (2). In the talk, I will explain the method used in this reference and show how it can be adapted to the study of (1) to obtain that, for any fixed T > 0, there exists a random field solution, and that this solution is unique and satisfies

sup(t,x)[0,T ]×Rd |u(t, x)| < , a.s.

This is ongoing joint work with A. Millet (U. Paris 1, Panthon-Sorbonne).


20/05/19: Alessandro Ramponi, Roma 2

Pricing valuation adjustments by correlation expansion

We consider firstly the problem of computing the Credit Value Adjustment (CVA) of a European option in a default intensity setting and in presence of the so-called Wrong Way Risk (WWR): that is, a decrease/increase in the credit quality of the counterpart produces a higher exposure in the portfolio of the derivative's holder. This effect may be modeled by the correlation between the stochastic factors driving our market model. We consider a method, introduced in the papers (F. Antonelli, A. Ramponi, S. Scarlatti, Review of Deriv. Research, 13 (2010)), which expands theoretically the solution to the PDE system in a Taylor's series with respect to the correlation parameters. Indeed, under quite general hypothesis, it is possible to verify that the solution to the PDE is regular with respect to the correlation parameters and therefore it can be expanded in series around the zero value for all of them. The coefficients of the series are characterized, by using Duhamel's principle, as solutions to a chain of PDE problems and they are therefore identified by means of Feynam-Kac formulas and expressed as expectations, that turn to be easier to compute or to approximate. Finally, we shoe that under appropriate conditions, the method can be extended to include several XVA's such as bilateral CVA, DVA, FVA and LVA due to collateralization. In fact, we remark that the adjusted value of a defaultable claim (with default risk of both parties) that takes into account the funding and collateralization costs verifies a (possibly nonlinear) BSDE and that, under some hypothesis, it may be approximated by using the correlation expansion method.


08/05/19: Aitor Muguruza, Imperial College

Smile properties of volatility derivatives: Unifying Malliavin calculus and Large Deviations Results

We analyze small-time implied volatility asymptotics for Realized Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle and Malliavin calculus. We will see that both approaches agree and complement each other in order to obtain a better understanding of small-time asymptotics. Based on our results, we develop a number of numerical schemes that efficiently compute small-time asymptotics and several examples will be given. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around the money. This talk is based on joint works with E. Alòs, C. Lacombe, D. Grcía-Lorite and H. Stone.

20/02/19: David Nualart UK

Rate of convergence in the Breuer-Major theorem via chaos expansions

In this talk we will present new estimates for the total variation and Wasserstein distances in the framework of the Breuer Major theorem. The results are based on the combination of Stein’s method and normal approximations together with Wiener chaos expansions.

13/02/19: Robert Dalang EPFL

On the density of the supremum of the solution to the linear stochastic heat equation

We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment of the solution over a (possibly degenerate) space-time rectangle. We establish a Gaussian-type upper bound on this density, which accounts for the size of the rectangle. This work is based on part of the Ph.D. thesis of my former student Fei Pu, and the preprint arXiv:1812.05310.

21/01/19: Lluis Quer-Sardanyons, UAB

Existence of density for the stochastic wave equation with space time homogeneous Gaussian noise

We consider the stochastic wave equation on R+ x R , driven by a linear multiplicative space time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by locally integrable functions. These functions are the Fourier transforms of tempered measures \nu and \mu, respectively. Under the assumption that \nu and \mu are absolutely continuous with respect to the Lebesgue measure, our main result shows that the law of the solution u(t,x) of this equation admits a density with respect to the Lebesgue measure.  The talk is based on joint work with Raluca Balan (University of Ottawa) and Jian Song (Shandong University).

19/12/18: Álvaro Leitao Rodriguez, UB
Shannon wavelets-based valuation of discretely monitored arithmetic Asian options

In this work, we propose an efficient and robust valuation of discretely monitored arithmetic Asian options bases on Shannon wavelets. We employ the so-called SWIFT method, a Fourier inversion numerical technique with respect to the existing related methods. Particularly interesting is that SWIFT provides mechanisms to determine all the free-parameters in the method, based on a prescribed precision in the density approximation. The method is applied to two general classes of dynamics: exponential Lévy models and square-root diffusions. Through the numerical experiments, we show that SWIFT outperforms state-of-the-art methods in terms of accuracy and robustness, and shows an impressive speed in execution time.

21/11/18: David Nualart, KU, USA

A central limit theorem for the stochastic heat equation

In this talk we present a central limit theorem for the space averages of the solution to a one- dimensional stochastic heat equation driven by a multiplicative space-time white noise. We obtain a rate of convergence in the total variation distance using Stein's method combined with Malliavin Calculus. We will also discuss a functional version of this central limit theorem.