Curs
2018-2019: Abstracts i Slides

Curs
2018-2019: Abstracts i Slides

**UPCOMING
SEMINARS**

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).

**PREVIOUS SEMINARS**

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

20/02/19: David Nualart UK

13/02/19: Robert Dalang EPFL

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

We consider the stochastic wave equation on R

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

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.