arXiv research

Reconstructs Riemannian geometry from diffusion properties.

The paper proves monotonicity formulas for solutions in Carnot groups, resembling well-known formulas for standard Laplacian and heat equations.

Riemannian metric matching learns the geometry of high-dimensional datasets using neural networks.

Consider a sample of $n$ points taken i.i.d from a submanifold $Σ$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $Σ$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carré du Champ for diffusion semi-groups, the theory of Empirical process…

In this paper, we obtain two rigidity results for $p$-Laplace type equation and $n$-Laplace equation with exponential nonlinearity on $n$-dimensional compact Riemannian manifolds by using of nonlinear flow and the carré du champ methods, respectively, where rigidity means that the PDE has only constant solution when a …

CDC-FM improves generative model quality-generalization tradeoff by regularizing with geometry-aware noise.

On a doubling metric measure space endowed with a "carré du champ", we consider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup and scale-invariant $L^p$ Poincaré inequalities $(P_p)$. We show that the combination of $(G_p)$ and $(P_p)$ for $p\ge 2$ always implies two-sided Gaussian heat kernel bounds. Th…

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter $ρ_1$, the generalized curvature inequality …

We formulate the Riemannian calculus of the probability set embedded with $L^2$-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability m…

Résumé. Nous définissons un groupoïde de Fréchet-Lie Map(S^1,X) d'ana-foncteurs du cercle vers un groupoïde de Lie X. Ceci fournit une présentation du Hom-champ Hom(S^1,\cX), où \cX est le champ différentiable associé à X. Nous appliquons cette construction au groupoïde de Lie sous-jacent au `gerbe fibré' d'une variété…

Let $(X,d,μ)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carré du champ". Assume that $(X,d,μ,\E)$ supports a scale-invariant $L^2$-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\i…

Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. Associated with $L$ one has \textit{le carré du champ} $Γ$ and a canonical distance $d$, with respect to which we suppose that …

On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to …

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singu…

In their seminal work Carr and Lee (2008) show how to robustly price and replicate a variety of claims written on the quadratic variation of a risky asset under the assumption that the asset's volatility process is independent of the Brownian motion that drives the asset's price. Additionally, they propose a correlatio…

Innovative advances validate a conjecture on maximal hypoellipticity in sub-Riemannian geometry.

Du, Kakade, Wang, and Yang recently established intriguing lower bounds on sample complexity, which suggest that reinforcement learning with a misspecified representation is intractable. Another line of work, which centers around a statistic called the eluder dimension, establishes tractability of problems similar to t…

In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} Δu$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the induced metric to establish two-sided bounds on the singular values of Du, which shows tha…

Study on 4-manifolds with positive scalar curvature.

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and unif…

Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most stochastic processes used in traditional option pricing models can be cast as special cases of time-changed Lévy processes. In particular these are models which can be tailored to exhibit correlated jumps in both the log price of assets and t…

Study symplectic cohomology of certain singularities using homological mirror symmetry.

The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.

New framework improves option pricing models by addressing volatility dynamics.

Gradient flow method solves isoperimetric inequality for maps.

We describe, in the general setting of closed cone fields, the set of causal functions which can be approximated by smooth Lyapunov. We derive several consequences on causality theory. Dans le contexte général des champs de cones fermés, on décrit l'ensemble des fonctions causales qui peuvent être approchées par des fo…

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

Smooth resolutions found for quotient of R^2 by infinite discrete groups.

The thesis examines stochastic calculus in option pricing with logistic models and numerical methods.

Let M be a smooth manifold, A a local algebra in sense of André Weil, M^{A} the manifold of near points on M of kind A and X(M^{A}) the module of vector fields on M^{A}. We give a new definition of vector fields on M^{A} and we show that X(M^{A}) is a Lie algebra over A. We study the cohomology of A-differential forms.

We compute the value of a variance swap when the underlying is modeled as a Markov process time changed by a Lévy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent Lévy measure, local stochastic volatility and have a local stochastic default intensity. Moreover, the Lévy subordina…

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

For any compact manifold of dimension n>=5, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian acting on diffential forms of degree 1<p<n-1 (exept for p=n/2 if n is even), within a given conformal class. When n<5 and when p=0,1,n-1,n, and p=n/2 if n is even, this simultaneous prescriptio…

We establish normal forms for conformal vector fields on pseudo-Riemannian manifolds in the neighborhood of a singularity. For real-analytic Lorentzian manifolds, we show that the vector field is analytically linearizable or the manifold is conformally flat. In either case, the vector field is locally conjugate to a no…

Let $M^n$ be a n-dimensional compact manifold, with $n\geq3$. For any conformal class C of riemannian metrics on M, we set $μ_k^c(M,C)=\inf_{g\in C}μ_{[\frac n2],k}(M,g)\Vol(M,g)^{\frac2n}$, where $μ_{p,k}(M,g)$ is the k-th eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that $0<μ_k^c(M,C)\leqμ_k^…

We introduce an algorithm for the pricing of finite expiry American options driven by Lévy processes. The idea is to tweak Carr's `Canadisation' method, cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted algorithm is viable for any Lévy process whose law at an independent, exponentially distrib…

This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded …

This is a short survey of Cheeger and Kleiner's nonembeddability theorem for Heisenberg group into $L^1$.

We improve the over-parametrization size over two beautiful results [Li and Liang' 2018] and [Du, Zhai, Poczos and Singh' 2019] in deep learning theory.

Minimal graphs grow slowly on curved spaces, proving constant solutions.

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan. In particular, we consider Merton…

We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector…

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their framework hinges on the stopping time property of the time changes. By analyzing the measurability of the time changes with respect to the underlying filtra…

The paper develops a new framework for pricing and hedging liquidity in crypto markets.

New method for pricing American options in time-dependent models, improving accuracy and efficiency.

A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connected…

We study the simply connected inextendable Lorentzian surfaces admitting a Killing vector field. We construct a natural family of such surfaces, that we call "universal extensions". They are characterized by a condition of symmetry, the "reflexivity", and a by a rather weak completeness assumption, the absence of "sadd…