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
θi,σi
∈
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).
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.
