arXiv research

Proves finite measure implies product structure for certain discrete subgroups.

The paper studies market viability and completeness in discrete markets.

This work relaxes OT problems with marginal moments constraints, achieving finite discrete measures.

The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.

Study shows $k$-NN classifier is not universally consistent on $(0,1)$ but consistent on discrete and specific measure spaces.

We give a complete characterization of both comonotone and not comonotone coherent risk measures in the discrete finite probability space, where each outcome is equally likely. To the best of our knowledge, this is the first work that characterizes \textit{and} distinguishes comonotone and not comonotone coherent risk …

We present sufficient conditions for topological stability of continuous functions $f:\mathbb{R}\to\mathbb{R}$ having finitely many local extrema with respect to averagings by discrete measures with finite supports.

We consider a discrete-time financial market model with finite time horizon and give conditions which guarantee the existence of an optimal strategy for the problem of maximizing expected terminal utility. Equivalent martingale measures are constructed using optimal strategies.

We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice emb…

Study loop ensembles on graphs, linking group theory and topology.

This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibl…

We study time-consistency questions for processes of monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a process of monetary risk measures time-consistent if it assigns to a process of financial values …

This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under non-dominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an opt…

Develops non-standard analysis for coherent risk estimation.

The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used…

Survey of mass partition problems in geometry and topology.

Defines speculative bubbles in discrete-time models based on discounted stock price losing mass.

Study shows singularity of stationary measure on Furstenberg boundary for certain random walks.

Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wassers…

The paper studies maps and reducibility for cocycles into CAT(0)-spaces.

New method for pricing financial products without no-arbitrage condition.

Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Ga…

Develops unbiased estimation method using underdamped Langevin dynamics.

Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then …

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric…

Novel analysis of EFP for finite-sum problems in neural networks.

New discrete-time model shows insider trading dynamics.

We introduce Network Maximal Correlation (NMC) as a multivariate measure of nonlinear association among random variables. NMC is defined via an optimization that infers transformations of variables by maximizing aggregate inner products between transformed variables. For finite discrete and jointly Gaussian random vari…

We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the push-forward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the Andrews-Curtis moves of the genera…

In this expository paper we illustrate the generality of game theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and the discrete pricing formulas…

A new method uses normalizing flows to approximate optimal transport between empirical distributions.

Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.

The study examines discrete subgroups of Lie groups and their residual finiteness.

Recently, Ross showed that it is possible to recover an objective measure from a risk-neutral measure. His model assumes that there is a finite-state Markov process X that drives the economy in discrete time. Many authors extended his model to a continuous-time setting with a Markov diffusion process X with state space…

We consider non-concave and non-smooth random utility functions with do- main of definition equal to the non-negative half-line. We use a dynamic pro- gramming framework together with measurable selection arguments to establish both the no-arbitrage condition characterization and the existence of an optimal portfolio i…

Study stationary measures and orbit closures for non-abelian actions on surfaces.

Estimates discontinuous optimal transport maps between a discrete and continuous distribution.

Framework for robust control under model uncertainty, improving financial derivatives hedging.

The article presents a general discrete time dividend valuation model when the dividend growth rate is a general continuous variable. The main assumption is that the dividend growth rate follows a discrete time semi-Markov chain with measurable space. The paper furnishes sufficient conditions that assure finiteness of …

The paper bounds payoffs and option prices in discrete models.

A method for vectorizing persistence diagrams simplifies topological data analysis.

The paper analyzes rates for a modified gradient descent method using Stein variational gradients.

In this paper a finite discrete time market with an arbitrary state space and bid-ask spreads is considered. The notion of an equivalent bid-ask martingale measure (EBAMM) is introduced and the fundamental theorem of asset pricing is proved using (EBAMM) as an equivalent condition for no-arbitrage. The Cox-Ross-Rubinst…

New finite element method for complex forms in any dimension.

In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which are explicitly constructed and shown to give a complete description of the enti…

Graphically discrete groups have strong rigidity properties.