Searcharxiv⌕ Search

arXiv subjects

Tongseok Lim

Publications and source records attributed to Tongseok Lim.

30 records · Page 2Linked to original sources
arXiv2022

Hodge theoretic reward allocation for generalized cooperative games on graphs

This paper generalizes L.S. Shapley's celebrated value allocation theory on coalition games by discovering and applying a fundamental connection between stochastic path integration driven by canonical time-reversible Markov chains and Hodge-theoretic discrete Poisson's equations on general weighted graphs. More precisely, we begin by defining cooperative games on general graphs and generalize Shapley's value allocation formula for those games in terms of stochastic path integral driven by the associated canonical Markov chain. We then show the value allocation operator, one for each player defined by the path integral, turns out to be the solution to the Poisson's equation defined via the combinatorial Hodge decomposition on general weighted graphs. Several motivational examples and applications are presented, in particular, a section is devoted to reinterpret and extend Nash's and Kohlberg and Neyman's solution concept for cooperative games. This and other examples, e.g. on revenue management, suggest that our general framework does not have to be restricted to cooperative games setup, but may apply to broader range of problems arising in economics, finance and other social and physical sciences.

math.PR↗
arXiv2021

Maximizing expected powers of the angle between pairs of points in projective space

Among probability measures on $d$-dimensional real projective space, one which maximizes the expected angle $\arccos(\frac{x}{|x|}\cdot \frac{y}{|y|})$ between independently drawn projective points $x$ and $y$ was conjectured to equidistribute its mass over the standard Euclidean basis $\{e_0,e_1,\ldots, e_d\}$ by Fejes Tóth \cite{FT59}. If true, this conjecture evidently implies the same measure maximizes the expectation of $\arccos^α(\frac{x}{|x|}\cdot \frac{y}{|y|})$ for any exponent $α> 1$. The kernel $\arccos^α(\frac{x}{|x|}\cdot \frac{y}{|y|})$ represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this {milder} conjecture in a non-empty range $α> α_{Δ^d} \ge 1$, and establish uniqueness of the resulting maximizer $\hat μ$ up to rotation. We show $\hat μ$ no longer maximizes when $α<α_{Δ^d}$. At the endpoint $α=α_{Δ^d}$ of this range, we show another maximizer $μ$ must also exist which is not a rotation of $\hat μ$. For the continuous version of the conjecture, an appendix provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield $α_{Δ^d}<2$. The original conjecture $\ald=1$ remains open (unless $d=1$). However, in the maximum possible range $α>1$, we show $\hat μ$ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the $L^\infty$-Kantorovich-Rubinstein-Wasserstein metric $d_\infty$ from optimal transportation; the same is true for any measure $μ$ which is mutually absolutely continuous with respect to $\hat μ$, but the size of the ball depends on {$α,d$, and} $\|\frac{d \hat μ}{dμ}\|_{\infty}$.

math.MG↗
arXiv2021

Isodiametry, variance, and regular simplices from particle interactions

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained -- apart from translations and rotations -- by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show these configurations are at least local energy minimizers in the relevant $d_\infty$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based on a simple isodiametric variance bound which characterizes regular simplices: it shows that among probability measures on ${\mathbf R}^n$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter) regular simplex.

math-ph↗
arXiv2021

On Fejes Tóth's conjectured maximizer for the sum of angles between lines

Choose $N$ unoriented lines through the origin of ${\bf R}^{d+1}$. The sum of the angles between these lines is conjectured to be maximized if the lines are distributed as evenly as possible amongst the coordinate axes of some orthonormal basis for ${\bf R}^{d+1}$. For $d \ge 2$ we embed the conjecture into a one-parameter family of problems, in which we seek to maximize the sum of the $α$-th power of the renormalized angles between the lines. We show the conjecture is equivalent to this same configuration becoming the {\em unique} optimizer (up to rotations) for all $α>1$. We establish both the asserted optimality and uniqueness in the limiting case $α=\infty$ of mildest repulsion. The same conclusions extend to $N=\infty$, provided we assume only finitely many of the lines are distinct.

math.MG↗
arXiv2020

Robust pricing and hedging of options on multiple assets and its numerics

We consider robust pricing and hedging for options written on multiple assets given market option prices for the individual assets. The resulting problem is called the multi-marginal martingale optimal transport problem. We propose two numerical methods to solve such problems: using discretisation and linear programming applied to the primal side and using penalisation and deep neural networks optimisation applied to the dual side. We prove convergence for our methods and compare their numerical performance. We show how adding further information about call option prices at additional maturities can be incorporated and narrows down the no-arbitrage pricing bounds. Finally, we obtain structural results for the case of the payoff given by a weighted sum of covariances between the assets.

math.PR↗
arXiv2020

Geometrical bounds for the variance and recentered moments

We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions. This is done using convex duality and (infinite-dimensional) linear programming. The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung's theorem (1901), and turns out to have applications to the aggregation dynamics modelling attractive-repulsive interactions: among probability measures on ${\mathbf R}^n$ whose support has diameter at most $\sqrt{2}$, we show that the variance around the mean is maximized precisely by those measures which assign mass $1/(n+1)$ to each vertex of a standard simplex. For $1 \le p <\infty$, the $p$-th moment --- optimally centered --- is maximized by the same measures among those satisfying the diameter constraint.

math.PR↗
arXiv2019

Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $λ-$Wasserstein metric topology with $1\le λ<\infty$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\infty$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.

math.PR↗
arXiv2019

Optimal Brownian stopping when the source and target are radially symmetric distributions

Given two probability measures $μ, ν$ on $\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $τ$ that maximize/minimize the cost functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $μ$ and with final distribution --once stopped at $τ$-- equal to $ν$. Under the assumption of radial symmetry on $μ$ and $ν$, we show that in dimension $d \geq 3$ and $α\neq 2$, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem to the optimal transportation problem for subharmonic martingales, and establish a duality result. This paper is an expanded version of a previously posted but not published work by the authors.

math.AP↗
arXiv2017

Optimal Brownian Stopping between radially symmetric marginals in general dimensions

Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim μ$ and with final distribution --once stopped at $τ$-- equal to $B_τ\sim ν$. The existence of such stopping times is guaranteed by Skorohod-type embeddings of probability measures in "subharmoic order" into Brownian motion. This problem is equivalent to an optimal mass transport problem with certain constraints, namely the optimal subharmonic martingale transport. Under the assumption of radial symmetry on $μ$ and $ν$, we show that the optimal stopping time is a hitting time of a suitable barrier, hence is non-randomized and is unique.

math.PR↗
arXiv2017

Dual attainment for the martingale transport problem

We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [BNT16] established such existence for weak (quasi-sure) duality, [BHP13] showed existence for the natural stronger pointwise duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when $y\mapsto c(x,y)$ is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost $c(x,y)$ is twice continuously differentiable in $y$. Further, this may not be improved as we give examples with $c(x,\cdot)\in C^{2-ε}$, $ε>0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.

math.PR↗
arXiv2016

Structure of optimal martingale transport plans in general dimensions

Given two probability measures $μ$ and $ν$ in "convex order" on $\R^d$, we study the profile of one-step martingale plans $π$ on $\R^d\times \R^d$ that optimize the expected value of the modulus of their increment among all martingales having $μ$ and $ν$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher dimensional case that we tackle in this paper. We show that many structural results can be obtained whenever a natural dual optimization problem is attained, provided the initial measure $μ$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $μ$-almost every $x$ in $\R^d$ is transported by the optimal martingale plan into a probability measure $π_x$ concentrated on the extreme points of the closed convex hull of its support. This will be established in full generality in the 2-dimensional case, and also for any $d\geq 3$ as long as the marginals are in "subharmonic order". In some cases, $π_x$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Many of the proofs rely on a remarkable decomposition of "martingale supporting" Borel subsets of $\R^d\times \R^d$ into a collection of mutually disjoint components by means of a "convex paving" of the source space. If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets. %On the other hand, they can also lead to natural and intriguing constructions of higher dimensional Nikodym sets.

math.AP↗