Minisymposium at the 5-th European Congress of Mathematics,


Weak Approximations of Stochastic Differential Equations


Amsterdam, July 14th-18th, 2008



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

Practical Matters:

The mini-symposium is scheduled during European Congress of Mathematics in the afternoon of 16th 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 London)




Mireille Bossy


Title of the talk:Discretization of non linear Langevin SDEs


Abstract: We consider a family of Langevin equationsarising in the Lagrangian approach for fluid mechanics models. Here we are interested by the numerical discretization of aLangevinsystem of SDEs(position and velocity) which are non-linearin the McKean sense, moreover the position must beconfined in a bounded domain and must take into accounta 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 xa, 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 σloc(t,f) depending on timet 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 Hurst parameter H>1/2.In particular, for the Euler scheme, the corresponding Multilevel estimator a root mean square error of order ε for the approximation of a functional of the terminal state of the SDE can be achieved with a computational effort of order ε-2.


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




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