Dear all,
  two presentations/seminars in Mathematics/Probability applied to Finance will be given by Profs Gilles Pagès (LPMA, Paris) and Abass Sagna (ENSIIE, Evry) next Wednesday, the 10th June 2015, from 11am to 13 am, in room 1BC45 in "Torre Archimede" (Dep. of Mathematics, University of Padova).

Titles and abstracts follow:

SEMINAR 1

Speaker 1: Prof. Gilles Pagès (LPMA, Paris)

Title : Greedy vector quantization (joint work with H. Luschgy)
Abstract (en Latex) :
We investigate the greedy version of the $L^p$-optimal vector quantization problem for an $\R^d$-valued random vector $X\!\in L^p$. We show the existence of a sequence $(a_N)_{N\ge 1}$ such that $a_N$ minimizes $a\mapsto\big \|\min_{1\le i\le N-1}|X-a_i|\wedge |X-a|\big\|_{L^p}$ ($L^p$-mean quantization error at level $N$ induced by $(a_1,\ldots,a_{N-1},a)$). We show that this sequence produces $L^p$-rate optimal $N$-tuples $a^{(N)}=(a_1,\ldots,a_{_N})$ ($i.e.$ the $L^p$-mean quantization error at level $N$ induced by $a^{(N)}$ goes to $0$ at rate $N^{-\frac 1d}$). Greedy optimal sequences also satisfy, under natural additional assumptions, the distortion mismatch property: the $N$-tuples $a^{(N)}$ remain rate optimal with respect to the $L^q$-norms, $p\le q <p+d$.
Finally, we propose optimization methods to compute greedy sequences, adapted from usual Lloyd's and Competitive Learning Vector Quantization procedures, either in their deterministic (implementable when $d=1$) for stochastic versions.


SEMINAR 2

Speaker 2: Prof. Abass Sagna (ENSIIE, Paris)

Title: Recursive Quantization of An Euler Diffusion Process and Its Application to Finance
Abstract (en Latex):
We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step $t_k$ of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times $t_0=0$ to time $t_k$. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method may sometimes be more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method.

Thanks for your attention,
Giorgia Callegaro