**Mireille Bossy (INRIA, Sophia Antipolis)
Emmanuel Gobet (Laboratoire Jean, Kuntzmann, Grenoble)
Peter Kloeden (Johann Wolfgang Goethe Universität,
Frankfurt)
Terry Lyons (Mathematical
Institute, Oxford)
**

**Practical Matters:**

** **

**The mini-symposium is scheduled during European Congress of Mathematics
in the afternoon of 16 ^{th} July 2008. There will be four talks of 40
minutes each: 13.25-14.05, 14.15-14.55, break, 15.25-16.05, 16.15-16.55. **

**Organizer:
Dan Crisan ( Imperial
College **

__Talks:__

**Mireille**** Bossy**

**Title of the talk: **Discretization of non linear Langevin SDEs

**Abstract:** We consider a family of Langevin
equations arising
in the Lagrangian approach for fluid mechanics
models. Here we are interested by the numerical discretization
of a Langevin
system of SDEs
(position and velocity) which are non-linear in the McKean sense, moreover the position
must be confined in a bounded domain and
must take into account a given velocity
at the boundary. Such numerical simulations are motivated by some
meteorological applications: we use such Lagrangian
model inside a computational cell of "classical meteorological
solver" in order to refine locally the wind computation. We will present
our numerical scheme for the Langevin system and we
will detail two of the main numerical difficulties. First, SDEs
have non-Lipschitz diffusion coefficients, typically
of the form x^{a}, 1/2≤a≤ 1.
Considering weak convergence, we give a rate of convergence result for a symmetrized scheme in a generic but 1D situation, without
confinement. Second, we propose a scheme for the confined process, but like for
reflected processes, the rate of weak convergence depends on the a priori regularity of solutions of the associated Kolmogorov PDE.

**Emmanuel
Gobet**

**Title of the talk: **Closed pricing formula via weak approximation of
financial models

**Abstract:** The standard Black-Scholes
formula (1973) has been derived under the assumption of lognormal diffusion
with constant volatility to price calls and puts. However those hypotheses are
unrealistic under real market conditions because we need to use different
volatilities to equate different option strikes *K* and maturities *T*.
Besides this, the market data shows that the shape of the implied volatilities
looks like a smile or a skew.

In order to fit
the smile or the skew, Dupire (1994) and Rubinstein
(1994) use a local volatility σ_{loc}(t,f) depending on time *t*
and state *f* to fit the market. This
hypothesis is interesting for hedging because it maintains the completeness of
the market.

However, only in
few cases, one has closed formulas.

But Andersen and
Andreasen (2000) show that this sole assumption of
local volatility is not compatible with empirical evidences (for instance, the
post crash of implied volatility for the S&P 500 index). Hence, they derived a model with local
volatility plus a jump process to fit the smile (we write it AA model). They
calibrate this model by solving the equivalent forward PIDE, but in the best
case, it leads to a time of calibration of the order of one minute.

In our work that
deals with weak approximation of models with local volatility and jumps, we
show how the stochastic analysis tools can be cleverly used to get remarkably
accurate formulas for the price of Call/Put options. The price is shown to be
equal to the (explicit) Merton's formula plus some (still explicit) correction
terms. This leads to a time of calibration smaller than one second.

This a joint
work with Eric BENHAMOU and Mohammed MIRI.

**Terry
Lyons**

**Title of the talk: **Resampling and Cubature on Wiener Space

**Abstract:** In some sense, the challenge of producing a weak
approximation to an SDE corresponds to asking for a good atomic description of
the law of the process (at some fixed time t).
The question is important in many contexts where one would like to solve
a parabolic PDE numerically. The work of
Kusuoka, Lyons and Victoir,
developing the method now called Cubature on Wiener Space, has demonstrated the
possibility of creating effective, high order particle methods, providing high
order approximations to these processes. However, these methods are not so
straightforward to implement, because of the tendency for the number of
particles to explode with the number of time steps. Further work by Litterer and the speaker
overcomes this obstacle, although it introduces quite a large constant into the
computational complexity of the algorithms (even if it reduces the order).

**Peter Kloeden**

**Title of the talk: **Convergence in stochastic numerics: some new developments

**Abstract:** Textbooks on numerical methods for stochastic
differential equations focus on strong and weak convergence and assume that all
necessary derivatives of the coefficients are globally bounded. In practice, however, calculations are done pathwise and the coefficients of most SDE and their
derivatives are not globally bounded. In the first part of this talk, I will
review some new developments on pathwise convergence
and indicate how the global boundedness assumptions
can be weakened. In the second part, I
will discuss the application of Giles' Multilevel Monte Carlo method to SDEs with additive fractional Brownian motion with ** ^{-2}**.

Joint work with Arnulf Jentzen,
Andreas Neuenkirch and Raffalella
Pavani.

Details
of how to register can be found at Registration
Information