arXiv research

Optimal reinsurance contracts designed for a continuum of risk types.

Generalizes risk sharing models to a continuum of agents.

We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and…

Study incentive efficiency in monopoly insurance markets with hidden information.

Gradient flows on graphons converge to curves on graphon space.

This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum …

Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constr…

Deriving option prices from operational-time Markov lattices

Study optimal portfolios for many players in a market model with random coefficients.

Smooth knots can be embedded into a specific Menger continuum.

The paper shows objective derivatives are covariant derivatives on Riemannian metrics.

Optimal risk sharing without convex preferences using aggregate convexity.

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq…

We introduce the continuum self-similar tree (CSST) and characterize it topologically. We apply this to answer a question of Curien about the topology of the continuum random tree (CRT). We also give a topological characterization of other trees with branch points of finite or infinite valences.

We consider the pricing and hedging of exotic options in a model-independent set-up using \emph{shortfall risk and quantiles}. We assume that the marginal distributions at certain times are given. This is tantamount to calibrating the model to call options with discrete set of maturities but a continuum of strikes. In …

Study on adaptivity to kernel regularity in bandit problems.

Continuum Dropout improves neural differential equations by preventing overfitting.

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary sup…

Derives continuum model from discrete $\varepsilon$-graphs with connectivity functional.

An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally f…

Generalizes Alexandroff's $V^n$-continua to cohomological dimensions.

Proves continuum limits of Lipschitz learning using Γ-convergence.

PCL framework optimizes climate risk management across three clusters.

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

A mesh-free method solves continuum-marginal optimal transport problems.

Optimal fuzzy classification aggregation functions are weighted means.

Generative model prices options and extracts risk-neutral densities.

A new model for heterogeneous populations optimizes consumption and investment over short horizons.

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…

This work proves the continuum limit of t-SNE for data visualization.

We consider the effect of recovery rates on a pool of credit assets. We allow the recovery rate to depend on the defaults in a general way. Using the theory of large deviations, we study the structure of losses in a pool consisting of a continuum of types. We derive the corresponding rate function and show that it has …

Study many-player investment-consumption games with power FPPs, finding market-risk preference affects consumption.

The paper analyzes optimal investment strategies in a game with jump risk, deriving mean field equilibria.

Many statistical learning problems can be posed as minimization of a sum of two convex functions, one typically a composition of non-smooth and linear functions. Examples include regression under structured sparsity assumptions. Popular algorithms for solving such problems, e.g., ADMM, often involve non-trivial optimiz…

Continuum transformers learn operators in context via gradient descent.

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and…

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…

A crucial challenge in image-based modeling of biomedical data is to identify trends and features that separate normality and pathology. In many cases, the morphology of the imaged object exhibits continuous change as it deviates from normality, and thus a generative model can be trained to model this morphological con…

It has been known for a long time that the fundamental group of the quotient of $\RR ^3$ by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.

Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano cont…

We analyze convergence of Fermat distances and their application in clustering.

Machine learning has witnessed tremendous success in solving tasks depending on a single hyperparameter. When considering simultaneously a finite number of tasks, multi-task learning enables one to account for the similarities of the tasks via appropriate regularizers. A step further consists of learning a continuum of…

Since its inception, the modus operandi of multi-task learning (MTL) has been to minimize the task-wise mean of the empirical risks. We introduce a generalized loss-compositional paradigm for MTL that includes a spectrum of formulations as a subfamily. One endpoint of this spectrum is minimax MTL: a new MTL formulation…

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic g…

Framework learns physics-informed continuum models from molecular data.