## Team : mathfi

## Section: New Results

**Keywords: ***quantization*.

## Quantization methods

**Participants:**G. Pagès, J. Printems.

The quantization method applied to mathematical finance or more
generally to systems of coupled stochastic differentials equations
(Forward/Backward) as introduced in [52] consists in the
approximation of the solution of the Backward Kolmogorov equation by
means of piecewise constant functions defined on an appropriate
Voronoï tesselation of the state space (R^{d}). The numerical
aspects of such a method are to compute such tesselations adapted to
the underlying diffusion and to estimate theirs transition
probabilities between different cells of two successive meshes
(after a time discretization procedure). Hence, it allows the
computation of a great number of conditional expectations along the
diffusions paths.

For these reasons, such a method seems to work well with the problem of valuation and hedging of financial products. More generally, its applications are concerned with the American options [52],[2], [16], stochastic control [25], nonlinear filtering and related problems (Zakai and McKean-Vlasov type stochastic partial differential equation [67]). See also [26] for a review on the subject.

There exists an ``infinite dimensional'' version of
the quantization method. In particular, when a stochastic process is
viewed as an Hilbert value random variable. It is the purpose of the
functional quantization to study such quantization (see [78]).
This new point of view allows us to treat the problem of valuation of
Asiatic options or more generally of ``path dependant'' options in an
L^{2} framework. See [32] for a numerical study for Asiatic
options and European options in the Heston stochastic volatility
model. In this paper, log-Romberg extrapolation in space seems to
provide very good results. A theoretical justification of such
numerical results should be a promising field of study. Note that this
should be implemented in the software premia (release 8) (see
Premia).

As concerned nonlinear filtering and quantization, let us note a different way of study. In a work in collaboration with Bruno Saussereau (Université de Franche-Comté, Besançon, France), we study the filtering of nonlinear systems from a numerical point of view: we want to compute the conditional expectations of signals when the observation process and/or the dynamic of the signal is not linear by means of a spectral approximation of the Zakai equation. In cite [77], the authors have proposed a spectral approach of nonlinear filtering by means of the Chaos expansion of the Wiener process. Numerical experiments based on this approach together with a quantization method provide promising results (see [40]).