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Open AccessArticle

Survival Probabilities for Correlated Drifted Brownian Motions via Exit from Simplicial Cones

by Tristan Guillaume

AppliedMath 2026, 6(3), 45; https://doi.org/10.3390/appliedmath6030045 - 10 Mar 2026

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This paper investigates the finite-horizon survival probability for a system of correlated arithmetic Brownian motions with heterogeneous drifts and volatilities, focusing on the event in which one component remains strictly below all others. Using a whitening transformation of the covariance structure, we reduce the problem to the survival of a standard Brownian motion in a simplicial cone, characterized by its spherical cross-section. While explicit solutions are available in low dimensions, we address the computationally challenging tetrahedral angular case. We derive a semi-analytic formula for the survival probability via an eigenfunction expansion of the Dirichlet Laplace–Beltrami operator on this curved domain. For efficient implementation, we construct a diffeomorphism from the spherical tetrahedron to a fixed Euclidean tetrahedron, enabling the computation of angular eigenpairs through a stable finite-element scheme. For higher-dimensional regimes, we also introduce a covariance-based difficulty index and geometric bounds based on an inscribed spherical cap to assess spectral convergence and estimate long-time decay rates. Numerical experiments show that this offline–online approach achieves high accuracy and substantial speedups relative to Monte Carlo benchmarks. Full article

(This article belongs to the Section Probabilistic & Statistical Mathematics)

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37 pages, 1282 KB

Open AccessArticle

Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins

by José Manuel García-Calcines, Luis Javier Hernández-Paricio and María Teresa Rivas-Rodríguez

Mathematics 2022, 10(4), 590; https://doi.org/10.3390/math10040590 - 14 Feb 2022

Viewed by 2495

Abstract

In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integers and by the join of augmented sequences. The cardinal functor that associates to each finite set X its cardinal

| X |

induces the sequential cardinal that associates to each augmented semi-simplicial finite set X an augmented sequence

{| X |}_{n}

of non-negative integers. We prove that the sequential cardinal functor is monoidal for the corresponding monoidal structures. This allows us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set as well as other augmented semi-simplicial sets which are built by joins. In this way, the monoidal structures of the augmented sequences of numbers may be thought of as an algebraization of the augmented semi-simplicial sets that allows us to do a simpler study of the combinatorics of the augmented semi-simplicial finite sets. Full article

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16 pages, 317 KB

Open AccessReview

From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications

by Octav Olteanu

Mathematics 2020, 8(8), 1328; https://doi.org/10.3390/math8081328 - 10 Aug 2020

Cited by 6 | Viewed by 2756

Abstract

The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of the type f ≤ h ≤ g, where f, −g are convex functionals and h is an affine functional, over a finite-simplicial set X, and proving a topological version for this result; (2) characterizing isotonicity of convex operators over arbitrary convex cones; giving a sharp direct proof for one of the generalizations of Hahn–Banach theorem applied to the isotonicity; (3) extending inequalities assumed to be valid on a small subset, to the entire positive cone of the domain space, via Krein–Milman or Carathéodory’s theorem. Thus, we point out some earlier, as well as new applications of the Hahn–Banach type theorems, emphasizing the topological versions of these applications. Full article

(This article belongs to the Special Issue Hahn-Banach Theorem, Polynomial Approximation, Moment Problems, and Related Inverse Problems)

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