arXiv subjects

Ting-Kam Leonard Wong

Publications and source records attributed to Ting-Kam Leonard Wong.

At least 19 recordsLinked to original sources

arXiv2026

Active Portfolio Management in Concentrated Equity Markets

The equal-weighted portfolio is a passive, rule-based strategy that has historically been difficult to outperform, delivering higher returns than the capitalization-weighted "market" benchmark across many markets and periods. Stochastic portfolio theory (SPT) reveals that this relative performance is regime dependent, with the equal-weighted portfolio underperforming during periods of increasing market concentration and high correlations, particularly market bubbles. These observations have motivated us to formulate and solve a stochastic control problem in which an investor actively allocates between the equal-weighted and market portfolios. The investor bases their allocation decisions on forecasts made under a flexible stochastic diversity--dispersion (SDD) model. Using a quadratic surrogate for implementation frictions, we characterize the optimal allocation through a linear forward--backward SDE and obtain an explicit "aiming in front of a moving target" representation of the optimal trading rate, in the spirit of Gârleanu and Pedersen. The penalty parameters are calibrated in sample to match the cumulative wealth effect of proportional transaction costs, while out-of-sample performance is evaluated with those costs deducted directly from portfolio wealth. Using historical S&P 500 data, we show that a mean-reverting SDD specification reproduces several empirical features of market diversity and dispersion. In out-of-sample backtests from 1995 to 2024, the resulting strategies deliver higher cumulative net returns than both the equal-weighted and market portfolios, and higher information ratios than the equal-weighted portfolio after 15-basis-point proportional transaction costs.

q-fin.PM↗

arXiv2026

Adapted Wasserstein Barycenters of Gaussian Processes

We study barycenters of filtered Gaussian processes in adapted Wasserstein space. The adapted Wasserstein distance refines classical optimal transport by requiring transport plans to respect the temporal flow of information, making it the natural metric for stochastic systems with filtration constraints, as in stochastic control, mathematical finance, and sequential decision problems. We prove that the \emph{unrestricted} barycenter problem for weighted Fréchet means of filtered Gaussian inputs admits a solution with Gaussian underlying law, representable as an enlarged filtered Gaussian process but not necessarily as an ordinary one. The problem decomposes into finitely many classical Bures--Wasserstein barycenter problems for the covariance contributions of the successive innovations. We then treat the \emph{restricted} problem, in which the barycenter is required to be an ordinary filtered Gaussian process, giving a rank and common-noise criterion for when the two problems agree, sufficient conditions for uniqueness, and first order optimality and regularity results. Under a martingale constraint we obtain an explicit solution via martingale projection and Bures--Wasserstein barycenters of the Gaussian increments. Beyond their intrinsic theoretical interest, our results provide a principled way to build representative models from collections of Gaussian stochastic systems, with applications to stochastic optimization, robust finance, and sequential statistical analysis.

math.PR↗

arXiv2026

On the Wasserstein alignment problem

Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces -- namely the source and the target measures -- the Wasserstein alignment problem seeks the transformation that minimizes the optimal transport cost between the pushforward of the source distribution and the target distribution, ensuring the closest possible alignment in a probabilistic sense. Examples of interest include two distributions on two Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^d$, and we want a spatial embedding of the $n$-dimensional source measure in $\mathbb{R}^d$ that is closest in some Wasserstein metric to the target distribution on $\mathbb{R}^d$. Similar data alignment problems also commonly arise in shape analysis and computer vision. In this paper, we show that this nonconvex optimal transport projection problem admits a convex Kantorovich-type dual that exploits statistical independence. This allows us to characterize the set of projections and devise a linear programming algorithm. For certain special examples, such as orthogonal transformations on Euclidean spaces of unequal dimensions and the $2$-Wasserstein cost, we characterize the covariance of the optimal projections. Our results also cover the generalization when we penalize each transformation by a function. An example is the inner product Gromov--Wasserstein distance minimization problem which has recently gained popularity.

math.PR↗

arXiv2026

A mathematical study of the excess growth rate

The excess growth rate, defined as the gap in Jensen's inequality for the logarithm, is a fundamental functional in portfolio theory. In this paper, we present a mathematical study motivated by information theory. We begin by establishing its properties and showing that it has rich connections with information theoretic concepts such as the Helmholtz free energy, L. Campbell's measure of average code length and large deviations. Our main results consist of three axiomatic characterization theorems of the excess growth rate, in terms of (i) the relative entropy, (ii) the gap in Jensen's inequality, and (iii) the logarithmic divergence that generalizes the Bregman divergence. Furthermore, we study maximization of the excess growth rate and compare it with the growth optimal portfolio. Our results not only provide theoretical justifications of the significance of the excess growth rate, but also establish new connections between information theory and quantitative finance.

cs.IT↗

arXiv2026

Adapted Optimal Transport between Filtered Gaussian Processes

We continue the study of adapted optimal transport in the discrete-time Gaussian setting. To this end, we introduce a space of filtered Gaussian processes where both the randomness and the flow of information are driven by a Gaussian white noise. On this space, the adapted $2$-Wasserstein distance (${AW}_2$) admits a variational representation as a constrained orthogonal Procrustes problem between Cholesky factors. Furthermore, the resulting quotient space is the ${AW}_2$-completion of the space of Gaussian distributions on the path space. We also characterize explicitly the ${AW}_2$-projections onto the subspaces of Gaussian martingales. Next, we analyze the adapted Brenier coupling -- a multivariate generalization of the Knothe--Rosenblatt coupling that serves as a myopic solution to the adapted transport problem, and compute its transport cost. Utilizing a Gaussian random matrix framework, we investigate the asymptotic behavior of transport costs as the time horizon grows; notably, we establish that the transport costs of all Gaussian bicausal couplings are asymptotically equivalent, whereas the classical Bures--Wasserstein distance is strictly smaller. Finally, we demonstrate that the adapted analogue of Gelbrich's lower bound fails in general, and we identify a sufficient martingale difference condition under which the bound is recovered.

math.PR↗

arXiv2026

Shape-constrained density estimation with Wasserstein projection

Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained density estimation via projection with respect to the $p$-Wasserstein distance, with a focus on the quadratic case $p = 2$. By considering shape constraints given by displacement convex subsets of the Wasserstein space, Wasserstein projection estimation is a convex optimization problem. We focus on two fundamental examples, namely non-increasing densities on $\mathbb{R}_+ := [0, \infty)$ and log-concave densities on $\mathbb{R}$. In each case, we prove structural properties of the Wasserstein projection estimator, propose a discretization which can be implemented by off-the-shelf solvers, and compare the projection estimator with the corresponding maximum likelihood estimator.

math.ST↗

arXiv2025

Maximum likelihood estimation for the $λ$-exponential family

The $λ$-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric point of view, but the development of computational methodologies is still in an early stage. In this paper, we propose a fixed point iteration for maximum likelihood estimation under i.i.d.~sampling, and prove using the duality that the likelihood is monotone along the iterations. We illustrate the algorithm with the $q$-Gaussian distribution and the Dirichlet perturbation.

math.ST↗

arXiv2025

Bregman-Wasserstein divergence: geometry and applications

The Bregman-Wasserstein divergence is the optimal transport cost when the underlying cost function is given by a Bregman divergence, and arises naturally in fields such as statistics and machine learning. We establish fundamental properties of the Bregman-Wasserstein divergence and propose a novel generalized transport geometry that promotes the Bregman geometry to the space of probability distributions. We provide a probabilistic interpretation involving exponential families and define generalized displacement interpolations compatible with the Bregman geometry. These interpolations are used to derive a generalized Pythagorean inequality, which is of independent interest. Furthermore, we construct a generalized dualistic geometry that lifts the differential geometry of the Bregman divergence to an infinite-dimensional statistical manifold. On the computational side, we demonstrate how Bregman-Wasserstein optimal transport maps can be estimated using neural approaches, establish the well-posedness of Bregman-Wasserstein barycenters, and relate them to Bayesian learning. Finally, we introduce the Bregman-Wasserstein JKO scheme for discretizing Riemannian Wasserstein gradient flows.

math.PR↗

arXiv2025

Macroscopic properties of equity markets: stylized facts and portfolio performance

Macroscopic properties of equity markets affect the performance of active equity strategies but many are not adequately captured by conventional models of financial mathematics and econometrics. Using the CRSP Database of the US equity market, we study empirically several macroscopic properties defined in terms of market capitalizations and returns, and highlight a list of stylized facts and open questions motivated in part by stochastic portfolio theory. Additionally, we present a systematic backtest of the diversity-weighted portfolio under various configurations and study its performance in relation to macroscopic quantities. All of our results can be replicated using codes made available on our online repository.

q-fin.ST↗

arXiv2025

Adapted optimal transport between Gaussian processes in discrete time

We derive explicitly the adapted $2$-Wasserstein distance between non-degenerate Gaussian distributions on $\mathbb{R}^N$ and characterize the optimal bicausal coupling(s). This leads to an adapted version of the Bures-Wasserstein distance on the space of positive definite matrices.

math.PR↗

arXiv2024

Efficient convex PCA with applications to Wasserstein GPCA and ranked data

Convex PCA, which was introduced in Bigot et al. (2017), modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this paper is threefold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.

stat.CO↗

arXiv2024

JKO schemes with general transport costs

We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges under suitable conditions to the corresponding Riemannian Fokker--Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker--Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence.

math.AP↗

arXiv2023

Information Geometry for the Working Information Theorist

Information geometry is a study of statistical manifolds, that is, spaces of probability distributions from a geometric perspective. Its classical information-theoretic applications relate to statistical concepts such as Fisher information, sufficient statistics, and efficient estimators. Today, information geometry has emerged as an interdisciplinary field that finds applications in diverse areas such as radar sensing, array signal processing, quantum physics, deep learning, and optimal transport. This article presents an overview of essential information geometry to initiate an information theorist, who may be unfamiliar with this exciting area of research. We explain the concepts of divergences on statistical manifolds, generalized notions of distances, orthogonality, and geodesics, thereby paving the way for concrete applications and novel theoretical investigations. We also highlight some recent information-geometric developments, which are of interest to the broader information theory community.

cs.IT↗

arXiv2022

Conformal Mirror Descent with Logarithmic Divergences

The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we introduce a generalization of continuous time mirror descent that we term the conformal mirror descent. We derive its dynamics under a generalized mirror map, and show that it is a time change of a corresponding Hessian gradient flow. We also prove convergence results in continuous time. We apply the conformal mirror descent to online estimation of a generalized exponential family, and construct a family of gradient flows on the unit simplex via the Dirichlet optimal transport problem.

math.OC↗

arXiv2022

Tsallis and Rényi deformations linked via a new $λ$-duality

Tsallis and Rényi entropies, which are monotone transformations of each other, are deformations of the celebrated Shannon entropy. Maximization of these deformed entropies, under suitable constraints, leads to the $q$-exponential family which has applications in non-extensive statistical physics, information theory and statistics. In previous information-geometric studies, the $q$-exponential family was analyzed using classical convex duality and Bregman divergence. In this paper, we show that a generalized $λ$-duality, where $λ= 1 - q$ is the constant information-geometric curvature, leads to a generalized exponential family which is essentially equivalent to the $q$-exponential family and has deep connections with Rényi entropy and optimal transport. Using this generalized convex duality and its associated logarithmic divergence, we show that our $λ$-exponential family satisfies properties that parallel and generalize those of the exponential family. Under our framework, the Rényi entropy and divergence arise naturally, and we give a new proof of the Tsallis/Rényi entropy maximizing property of the $q$-exponential family. We also introduce a $λ$-mixture family which may be regarded as the dual of the $λ$-exponential family, and connect it with other mixture-type families. Finally, we discuss a duality between the $λ$-exponential family and the $λ$-logarithmic divergence, and study its statistical consequences.

math.PR↗

arXiv2021

Functional portfolio optimization in stochastic portfolio theory

In this paper we develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets, and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the generating function. We prove an existence and uniqueness result for our optimization problem and provide a stability estimate in terms of a Wasserstein metric of the input measure. Then, we formulate a discretization which can be implemented numerically using available software packages and analyze its approximation error. Finally, we present empirical examples using CRSP data from the US stock market, including the performance of the portfolios allowing for dividends, defaults, and transaction costs.

q-fin.PM↗

arXiv2021

Random concave functions

Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gradient maps define novel investment strategies. The gradient maps may also be regarded as optimal transport maps on the simplex. In this paper we construct and study probability measures supported on spaces of concave functions. These measures may serve as prior distributions in Bayesian statistics and Cover's universal portfolio, and induce distribution-valued random variables via optimal transport. The random concave functions are constructed on the unit simplex by taking a suitably scaled (mollified, or soft) minimum of random hyperplanes. Depending on the regime of the parameters, we show that as the number of hyperplanes tends to infinity there are several possible limiting behaviors. In particular, there is a transition from a deterministic almost sure limit to a non-trivial limiting distribution that can be characterized using convex duality and Poisson point processes.

math.PR↗

arXiv2021

Projections with logarithmic divergences

In information geometry, generalized exponential families and statistical manifolds with curvature are under active investigation in recent years. In this paper we consider the statistical manifold induced by a logarithmic $L^{(α)}$-divergence which generalizes the Bregman divergence. It is known that such a manifold is dually projectively flat with constant negative sectional curvature, and is closely related to the $\mathcal{F}^{(α)}$-family, a generalized exponential family introduced by the second author. Our main result constructs a dual foliation of the statistical manifold, i.e., an orthogonal decomposition consisting of primal and dual autoparallel submanifolds. This decomposition, which can be naturally interpreted in terms of primal and dual projections with respect to the logarithmic divergence, extends the dual foliation of a dually flat manifold studied by Amari. As an application, we formulate a new $L^{(α)}$-PCA problem which generalizes the exponential family PCA.

math.DG↗