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Section: New Results
Numerical Probability
American option pricing.
Damien Lamberton with M. Pistorius has worked on the approximation of American options by Canadian options, which originated from the work of Peter Carr. This lead them to revise old results on the binomial approximation of the American put. D. Lamberton is also working with M. Zervos on American options involving the maximum of the underlying.
Convergence in total variation of approximation schemes for Markov processes
(V. Bally and PhD student C. Rey [40] )
The main issue was to consider very general approximation schemes and to estimate the approximation error for test functions which are just measurable and bounded. It is worth to mention that the input of noise in the approximation schemes is allowed to be quite general, while in the standard approximation schemes for diffusion processes one considers Gaussian input only. In some sense this means that we treat invariance principles as well. We also considered approximation schemes of higher order, as the Victoir Nynomia scheme for example. An important ingredient is an abstract Malliavin calculus for general random variables (which has been settled in previous papers of V. Bally and Lucia Caramellino.
Approximation schemes for Piecewise Deterministic Markov Processes
(V. Bally and PhD student V.Rabiet [39] ).
PDMP processes are very popular in many practical fields as biology, chemistry or fiability theory. The main idea is that such a model may present different scales: slow ones and rapid onces. And from a numerical point of view it is extremely difficult to implement algorithms which take care of rapid scales in details. Then the idea is to average the rapid scales (in the spirit of the Central Limit Theorem) and consequently to replace small (and rapid) jumps by a Brownien component. This procedure is already widely used by practitioners. Our work was to derive estimates of the error which is done by this procedure.
Convergence in distribution norms in the Central Limit Theorem
(V. Bally with Lucia Caramellino and Guillaume Poly)
In the classical theory, the convergence which has already been studied is the convergence in total variation (measurable test functions). The main result is the theorem of Prohorov, in the fifties. We have proved that under similar hypothesis (with more finite moments however) one may obtain a much more accurate estimate of the error, in some norms which are close to distribution norms. As a remarkable consequence, we obtained a CLT for the zeros of trigonometric polynomials with random coefficients.