## Welcome to Tongseok Lim’s homepage!

Tongseok Lim is an assistant professor in Quantitative Methods, Mitchell E. Daniels, Jr. School of Business at Purdue University.

Tongseok Lim is interested in various fields, and his study includes (i) (Martingale–) Optimal Transport in multi-dimensions and its applications to Economics, Finance and Statistics, (ii) Analysis of Variational problems arising in Physics, Geometry and Data Science, and (iii) Hodge Theory on graphs and its connection with Stochastic Calculus and Game Theory.

Please see CV and Google Scholar for more information about Tongseok Lim.

⁍ News (Jun 2024): Lim’s paper, announced in Mar 2022, has been completely rewritten with new results and a new title: Cooperative networks and Hodge-Shapley value.

⁍ News (Nov 2023): Lim’s new paper, Optimal exercise decision of American options under model uncertainty, has now appeared. When the model uncertainty consistent with the given marginal distribution information is described as the martingale optimal transport problem, the paper shows that any option exercise scheme associated with any market model that jointly maximizes the expected option payoff must be nonrandomized if the American option payoff satisfies a suitable convexity condition and the model-free price upper bound and its relaxed version coincide. The latter condition is desired to be removed under appropriate conditions on the cost and marginals.

⁍ News (Oct 2023): Lim’s new paper, Node classification in networks via simplicial interactions, has now appeared. The paper proposes a probability-based objective function for semi-supervised network node classification that takes advantage of higher-order networks’ capabilities. We evaluate the function using both balanced and imbalanced datasets generated by the Planted Partition Model (PPM), as well as a real-world political book dataset. According to the results, in challenging classification contexts, higher-order networks outperform pairwise interactions in terms of objective function performance. This is a collaboration with Eunho Koo, a postdoctoral researcher at KIAS Center for AI.

⁍ News (Sep 2023): Lim’s new paper, Geometry of vectorial martingale optimal transport and robust option pricing, has now appeared. The paper investigates the Vectorial Martingale Optimal Transport (VMOT) problem, the geometry of its solutions, and its application with robust option pricing problems in finance. The findings demonstrate a subtle relationship between spatial dimension, cost function supermodularity, and their effect on the geometry of solutions to the VMOT problem. Applications of the model to financial problems demonstrate how the dimensional reduction caused by monotonicity can be used to improve existing computational methods. This is joint work with Brendan Pass and two PhD students we jointly supervise: Joshua Hiew and Marcelo Souza.

⁍ News (Jul 2023): Lim’s new paper, Replication of financial derivatives under extreme market models given marginals, has now appeared. The paper shows that a portfolio sub- or super-hedging a general path-dependent derivative security in terms of trading European options and underlyings exists, and furthermore, the portfolio replicates the derivative payoff when the market model yields the extremal price of the derivative given marginal distributions of the underlyings. Mathematically, the paper establishes dual attainment for the multi-period vectorial martingale optimal transport problem.

⁍ News (May 2023): Lim’s paper, Maximal monotonicity and cyclic involutivity of multiconjugate convex functions, will be published in SIAM Journal on Optimization.

⁍ News (Mar 2023): Lim’s paper, Geometry of vectorial martingale optimal transportations and duality, will be published in Mathematical Programming series A. The paper investigates duality and its strong attainment of the martingale optimal transport problem given vectorial marginal distributions, which is motivated by robust mathematical finance.

⁍ News (Dec 2022): Lim’s paper, announced in July 2022, has been significantly expanded from the previous version, and bears a new title: Maximal monotonicity and cyclic involutivity of multiconjugate convex functions.

⁍ News (Nov 2022): Lim’s paper, Classifying minimum energy states for interacting particles: Regular simplices, will be published in Communications in Mathematical Physics. This is joint work with Robert J. McCann and his student Cameron Davis.

⁍ News (Jul 2022): Lim’s new paper, Completeness and maximal monotonicity of multi-conjugate convex functions on the line, has now appeared. The paper introduces the notion of generalized involution of several convex functions which we call completeness of convex conjugation, and studies its validity and connection with maximal monotonicity induced by multi-conjugate convex functions.

⁍ News (May 2022): Lim’s paper, Classifying minimum energy states for interacting particles: Spherical shells, will be published in SIAM Journal on Applied Mathematics. This is joint work with Robert J. McCann and his student Cameron Davis.

⁍ News (Mar 2022): Lim’s presentation was given at the BIRS workshop Video, explaining the paper Stochastic integral representation of solutions to Hodge theoretic Poisson’s equations on graphs, and cooperative value allocation of Shapley and Nash. I hope you enjoy it! The paper combines Lim’s preceding articles in the News (Jul 2021).

⁍ News (Feb 2022): Lim’s paper, On the cardinality of sets in R^d obeying a slightly obtuse angle bound, will be published in SIAM Journal on Discrete Mathematics. This is joint work with Robert J. McCann.

⁍ News (Jan 2022): Lim’s paper, Maximizing expected powers of the angle between pairs of points in projective space, has been published online in Probability Theory and Related Fields. This is joint work with Robert J. McCann.

⁍ News (Sep 2021): Lim’s new papers, Classifying minimum energy states for interacting particles: Spherical Shells, and Classifying minimum energy states for interacting particles: Regular Simplices, have now appeared. These are joint work with Robert J. McCann and his student Cameron Davis. The papers study the interaction energy minimization problem in which the energy is governed by a power-law, borderline mildly-repulsive potential. We provide a complete characterization of the minimizers for this critical regime and beyond.

⁍ News (Jul 2021): Lim’s new papers, A Hodge theoretic extension of Shapley axioms and Hodge theoretic reward allocation for generalized cooperative games on graphs, have now appeared. The paper proposes a new extension of Shapley’s celebrated axioms for cooperative games in light of Hodge decomposition, and realize them through an intriguing stochastic path integral driven by a canonical, time-reversible Markov chain. This paper was inspired by the nice work of A. Stern and A. Tettenhorst.