No Arabic abstract
Pitmans theorem states that if {Bt, t $ge$ 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf s$le$t Bs, t $ge$ 0} is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob sense to remain forever positive. In this paper one gives a similar representation for the Brownian motion in an interval. Due to the double barrier, it is much more involved and only asymptotic. This uses the fact that the interval is an alcove of the Affine Lie algebra A 1 1 .
We present some results about connections between Littelmann paths and Brownian paths in the framework of affine Lie algebras. We expect that they will be the first steps on a way which could hopefully lead to a Pitman type theorem for a Brownian motion in an alcove associated to an affine Weyl group.
We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra $mathfrak{g}$ considering tensor product of irreducible highest weight modules of $mathfrak{g}$ and specializations of the characters involving the Weyl vector $rho$. We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of $mathfrak{g}$.
This is a summary (in French) of my work about brownian motion and Kac-Moody algebras during the last seven years, presented towards the Habilitation degree.
In this paper, we study rough path properties of stochastic integrals of It^{o}s type and Stratonovichs type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.
In some non-regular statistical estimation problems, the limiting likelihood processes are functionals of fractional Brownian motion (fBm) with Hursts parameter H; 0 < H <=? 1. In this paper we present several analytical and numerical results on the moments of Pitman estimators represented in the form of integral functionals of fBm. We also provide Monte Carlo simulation results for variances of Pitman and asymptotic maximum likelihood estimators.