arXiv research

Upper bound on expected supremum of Bernoulli process.

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

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

The paper evaluates functions of stable Lévy processes and their extrema efficiently.

Study vector-valued robust control under uncertainty.

The paper establishes conditions for Bayesian consistency in supremum metric.

In this paper we extend the stability results of [4]}. Our utility maximization problem is defined as an essential supremum of conditional expectations of the terminal values of wealth processes, conditioned on the filtration at the stopping time $τ$. To establish our results, we extend the classical results of convex …

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…

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…

A new framework tightens risk measure confidence bounds.

We propose a robust risk measurement approach that minimizes the expectation of overestimation plus underestimation costs. We consider uncertainty by taking the supremum over a collection of probability measures, relating our approach to dual sets in the representation of coherent risk measures. We provide results that…

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 …

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.

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…

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

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

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a …

Paper relaxes triangle inequality for KL divergence between Gaussian distributions.

Unified framework for information-theoretic bounds on learning algorithms.

We study the optimal investment problem for a continuous time incomplete market model such that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are assumed to be independent from the driving Brownian motion, and they are supposed to be currently observable. It is shown t…

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.

New risk measures adjust for tail risk inadequacies.

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 calculates extreme measures in continuous time conic finance.

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows…

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

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

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programm…

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…

In a discrete-time financial market, a generalized duality is established for model-free superhedging, given marginal distributions of the underlying asset. Contrary to prior studies, we do not require contingent claims to be upper semicontinuous, allowing for upper semi-analytic ones. The generalized duality stipulate…

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.

Consider an agent who enters a financial market on day t = 0 with an initial capital amount x. He invests this amount on stocks and the money market, and by day t = T, has generated a wealth W . He is given a convex class of probability measures (called scenarios) and a real-valued function (or floors) corresponding to…

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

Non-asymptotic tail bounds for Kostlan-Shub-Smale field on sphere

The study analyzes the performance of a nonparametric estimator for dynamical systems.