arXiv research

The paper establishes conditions for Bayesian consistency in supremum metric.

Develops a new essential supremum concept for financial models.

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

We show how to compute lower bounds for the supremum Bayes error if the class-conditional distributions must satisfy moment constraints, where the supremum is with respect to the unknown class-conditional distributions. Our approach makes use of Curto and Fialkow's solutions for the truncated moment problem. The lower …

Upper bound on expected supremum of Bernoulli process.

We consider the optimal prediction problem of stopping a spectrally negative Lévy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We…

Study optimal control of diffusion processes with infimum or supremum costs.

An elementary proof shows submodular functions can be represented as measure suprema.

Paper relaxes triangle inequality for KL divergence between Gaussian distributions.

The {\em drawdown} process $Y$ of a completely asymmetric Lévy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the Lévy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the leve…

Study short-term behavior of up-and-in barrier options using Malliavin calculus.

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding Kähler-Einstein metrics. We show that on P^2 blown up in one point this sup…

Derives integral representations for a Lévy process and its extremum, hitting time, with fast evaluation.

This paper considers the valuation of exotic path-dependent options in Lévy models, in particular options on the supremum and the infimum of the asset price process. Using the Wiener--Hopf factorization, we derive expressions for the analytically extended characteristic function of the supremum and the infimum of a Lév…

Logistic regression for brain imaging without p-values.

In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $λ(\cdot)$ for Kähler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $λ(\cdot)$ in Kähler class $2πc_1(M)$ under an assumption that the modified Mabuchi's K-energy $μ(\cdot)$ defined …

The paper constructs optimal confidence bands for kernel gradient flow estimators.

Study on stable translation lengths of surface homeomorphisms and their approximations.

In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specifie…

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We …

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the 1-skeleton. The theorem is proved by applying a sort of volume-increasing flow to any h…

Vertex distortion measures how far lattice knots deviate from straight lines.

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_ζ(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always g…

Develops a Monte Carlo algorithm for tempered stable process extrema.

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

The paper proves a Moser-Trudinger inequality for zero-mean functions in 2D.

Given a finite honest time, we first show that the associated Azéma optional supermartingale can be expressed as the drawdown and the relative drawdown of some local optional supermartingales with continuous running supremum. The relative drawdown representation then allows us to provide a characterisation of finite ho…

Study heat profiles and eigenfunctions using Brownian motion.

Gaussian process (GP) regression is a powerful interpolation technique due to its flexibility in capturing non-linearity. In this paper, we provide a general framework for understanding the frequentist coverage of point-wise and simultaneous Bayesian credible sets in GP regression. As an intermediate result, we develop…

Let $M$ be a compact manifold of dimension $n$. In this paper, we introduce the {\em Mass Function} $a \geq 0 \mapsto \xp{M}{a}$ (resp. $a \geq 0 \mapsto \xm{M}{a}$) which is defined as the supremum (resp. infimum) of the masses of all metrics on $M$ whose Yamabe constant is larger than $a$ and which are flat on a ball…

Designs efficient algorithms to maximize the expectation of Gaussian random variables.

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a…

We provide a model-free pricing-hedging duality in continuous time. For a frictionless market consisting of $d$ risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path dependent European option has the same value as the purely probabi…

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The technique is: minimal surfaces, Thurston norm in homology and new conformal invari…

We study the pointwise supremum of convex integral functionals $\mathcal{I}_{f,γ}(ξ)= \sup_{Q} \left( \int_Ωf(ω,ξ(ω))Q(dω)-γ(Q)\right)$ on $L^\infty(Ω,\mathcal{F},\mathbb{P})$ where $f:Ω\times\mathbb{R}\rightarrow\overline{\mathbb{R}}$ is a proper normal convex integrand, $γ$ is a proper convex function on the set of p…

The purpose of this paper is to point out that `supremum' in two inequalities of Brooks should be replaced with `infimum'. The results of this paper are already known by Professor Higuci. Hence, I want to delete this paper from this preprint server.

Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremum of Lévy processes. In particular, we extend the results of Broadie et al. (1999) to jump-diffusion models. We also derive bounds for general exponential Lévy models.

A new framework tightens risk measure confidence bounds.

The paper bounds the expectation of empirical processes indexed by Hölder classes.

This paper develops Yang-Mills flow on Riemannian manifolds with special holonomy. By analogy with the second-named author's thesis, we find that a supremum bound on a certain curvature component is sufficient to rule out finite-time singularities. Assuming such a bound, we prove that the infinite-time bubbling set is …

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic ave…

Improved bounds for Monte Carlo Rademacher Averages using self-bounding functions.

Characterizes super-replication prices in a financial market model.

New method for distributional off-policy evaluation using Bellman residual minimization.

We study an equivalence of (i) deterministic pathwise statements appearing in the online learning literature (termed \emph{regret bounds}), (ii) high-probability tail bounds for the supremum of a collection of martingales (of a specific form arising from uniform laws of large numbers for martingales), and (iii) in-expe…

Study vector-valued robust control under uncertainty.

We prove that, given $|H|<1$, a generic simple closed curve embedded in the asymptotic boundary of $\mathbb{H}^3$ (with respect to the supremum metric) bounds more than one complete surface embedded in $\mathbb{H}^3$ which has constant mean curvature $H$. We remark that this is not true for the space of simple closed $…

Method simulates drawdown and duration in Lévy models using Gaussian approximation.