Papers by Fabrice Baudoin
arXiv preprint arXiv:0806.2547, Jun 16, 2008
Abstract: We describe three elementary models in three dimensional subelliptic geometry which cor... more Abstract: We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU (2), Heisenberg and SL (2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the $\ Gamma_ {2} $ techniques that we adapt to those elementary model spaces. The important feature developed here is that although the ...
Page 1. Elect. Comm. in Probab.
We remind that a stochastic process (Xt) t≥ 0 defined on a probability space (Ω, F, P) is said to... more We remind that a stochastic process (Xt) t≥ 0 defined on a probability space (Ω, F, P) is said to be a Gaussian process if for every t1,···, tn∈ R≥ 0, the random vector (Xt1,···, Xtn) is Gaussian. The distribution of a Gaussian process (Xt) t≥ 0 is uniquely determined by its mean function m (t)= E (Xt), and its covariance function
Abstract. We extend to Riemannian manifolds the theory of conditioned stochastic differential equ... more Abstract. We extend to Riemannian manifolds the theory of conditioned stochastic differential equations (see [1]). We also provide some enlargement formulas of the Brownian filtration in this non-flat setting. When it is possible, we construct by means of the horizontal Bismut-Malliavin's calculus of variations some martingales in an enlarged filtration which are the analogues of the Newton's martingales introduced in [4].
The results of [4] are extended under weaker assumptions to d-dimensional and possibly discontinu... more The results of [4] are extended under weaker assumptions to d-dimensional and possibly discontinuous processes and applied to the modelling of weak anticipations both on complete and incomplete financial markets. In the case of a complete market, we show that there exists a minimal probability measure associated with an anticipation. Remarkably, this minimal probability does not depend on the selected utility function. Throughout the paper, Markovian models are studied in details as canonical examples.
The first stochastic process that has been extensively studied is the so-called Brownian motion, ... more The first stochastic process that has been extensively studied is the so-called Brownian motion, named in honor of the botanist Robert Brown (1773-1858), who observed and described in 1828 the random movement of particles suspended in a liquid or gas. One of the first mathematical studies of this process goes back to the mathematician Louis Bachelier (1870-1946), in 1900, who presented a stochastic modelling of the stock and option markets.
We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion ... more We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations.
For a given functional Y on the path space, we define the pinning class of the Wiener measure as ... more For a given functional Y on the path space, we define the pinning class of the Wiener measure as the class of probabilities which admit the same conditioning given Y as the Wiener measure. Using stochastic analysis and the theory of initial enlargement of filtration, we study the transformations (not necessarily adapted) which preserve this class. We prove, in this non Markov setting, a stochastic Newton equation and a stochastic Noether theorem.
Abstract: We describe three elementary models in three dimensional subelliptic geometry which cor... more Abstract: We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU (2), Heisenberg and SL (2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the $\ Gamma_ {2} $ techniques that we adapt to those elementary model spaces.
The aim of the present survey is to give an outline of the modern mathematical tools which can be... more The aim of the present survey is to give an outline of the modern mathematical tools which can be used on a financial market by a” small” investor who possesses some information on the price process. Financial markets obviously have asymmetry of information. That is, there are different types of traders whose behavior is induced by different types of information that they possess. Let us consider a” small” investor who trades in an arbitrage free financial market so as to maximize the expected utility of his wealth at a given time horizon.
Abstract These notes focus on the applications of the stochastic Taylor expansion of solutions of... more Abstract These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern-Gauss-Bonnet theorem.
Abstract: We study the heat kernel of the sub-Laplacian L on the CR sphere S2n+ 1. An explicit an... more Abstract: We study the heat kernel of the sub-Laplacian L on the CR sphere S2n+ 1. An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian-L+ n2 that was obtained by Geller [12], and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration S2n+ 1\ rightarrow CPn.
Abstract: We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Li... more Abstract: We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $ L^ p $-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in\ cite {BaudoinGarofalo2011}.
Abstract We study the two-dimensional fractional Brownian motion with Hurst parameter H> ½. In pa... more Abstract We study the two-dimensional fractional Brownian motion with Hurst parameter H> ½. In particular, we show, using stochastic calculus, that this process admits a skew-product decomposition and deduce from this representation some asymptotic properties of the motion.
Abstract The purpose of this work is to study some monotone functionals of the heat kernel on a c... more Abstract The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau (Acta Math. 156, 153–201, 1986), the gradient estimate of Ni (J. Geom. Anal.
Abstract: Let $\ M $ be a smooth connected manifold endowed with a smooth measure $\ mu $ and a s... more Abstract: Let $\ M $ be a smooth connected manifold endowed with a smooth measure $\ mu $ and a smooth locally subelliptic diffusion operator $ L $ satisfying $ L1= 0$, and which is symmetric with respect to $\ mu $. Associated with $ L $ one has\ textit {le carr\'e du champ} $\ Gamma $ and a canonical distance $ d $, with respect to which we suppose that $(M, d) $ be complete.
Abstract: In this paper, we seek to measure the impact of arbitrage opportunities on some key res... more Abstract: In this paper, we seek to measure the impact of arbitrage opportunities on some key results of asset pricing theory. We do this by showing how a market that is viable, but potentially badly arbitraged, can exist if the agents' preferences is modified. In this case, the pricing measure is not necessarily positive and can no longer be treated as a risk-neutral probability. Unless such arbitrage opportunities are of the cash-and-carry kind, prices remain martingale under this potentially negative measure.
Abstract. In this paper we consider an investor who trades in a complete financial market so as t... more Abstract. In this paper we consider an investor who trades in a complete financial market so as to maximize his expected utility of wealth at a prespecified time. We assume that he is in the following position: His portfolio decisions are based on a public information flow but he possesses extra information about the law of some functional of the future prices of a stock. Our basic question is then: How should he trade on the financial market to optimally exploit his extra information?
The goal of this article is to understand geometrically the asymptotic expansion of stochastic fl... more The goal of this article is to understand geometrically the asymptotic expansion of stochastic flows. Precisely, we show that a hypoelliptic diffusion can be pathwise approximated at each (normal) point by the lift of a Brownian motion in a graded nilpotent group with dilations. This group, called a Carnot group, appears as a tangent space in Gromov-Hausdorff s sense. We then apply this geometrical point of view in different domains:• the study of the spectrum of regular sub-Laplacians;• the study of Riemannian Brownian motions.
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Papers by Fabrice Baudoin