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76 pages, 654 KiB

Open AccessArticle

Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold

by Qi Feng and Wuchen Li

Entropy 2023, 25(5), 786; https://doi.org/10.3390/e25050786 - 11 May 2023

Cited by 3 | Viewed by 2116

We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric. Full article

(This article belongs to the Special Issue Information Geometry and Its Applications)

13 pages, 738 KiB

Open AccessArticle

On Deterministic and Stochastic Multiple Pathogen Epidemic Models

by Fernando Vadillo

Epidemiologia 2021, 2(3), 325-337; https://doi.org/10.3390/epidemiologia2030025 - 12 Aug 2021

Cited by 1 | Viewed by 3103

Abstract

In this paper, we consider a stochastic epidemic model with two pathogens. In order to analyze the coexistence of two pathogens, we compute numerically the expectation time until extinction (the mean persistence time), which satisfies a stationary partial differential equation with degenerate variable coefficients, related to backward Kolmogorov equation. I use the finite element method in order to solve this equation, and we implement it in FreeFem++. The main conclusion of this paper is that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two diseases and in the extinction outcome of one of them. Now, the main challenge would be to find an explanation for this result. Full article

(This article belongs to the Special Issue Mathematical Modeling for Understanding Viral Infections Within-Host and Between-Host)

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18 pages, 585 KiB

Open AccessEditor’s ChoiceArticle

A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

by Stefan Kremsner, Alexander Steinicke and Michaela Szölgyenyi

Risks 2020, 8(4), 136; https://doi.org/10.3390/risks8040136 - 9 Dec 2020

Cited by 16 | Viewed by 4804

Abstract

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks. Full article

(This article belongs to the Special Issue Computational Finance and Risk Analysis in Insurance)

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33 pages, 467 KiB

Open AccessArticle

Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

by Haesung Lee and Gerald Trutnau

Symmetry 2020, 12(4), 570; https://doi.org/10.3390/sym12040570 - 5 Apr 2020

Cited by 3 | Viewed by 2824

Abstract

We show uniqueness in law for a general class of stochastic differential equations in

R^{d}

d \geq 2

, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy [...] Read more.

We show uniqueness in law for a general class of stochastic differential equations in

R^{d}

d \geq 2

, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Points of degeneracy have a d-dimensional Lebesgue–Borel measure zero. Weak existence is obtained for a more general, but not necessarily locally bounded drift coefficient. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

16 pages, 2378 KiB

Open AccessArticle

Computational Principle and Performance Evaluation of Coherent Ising Machine Based on Degenerate Optical Parametric Oscillator Network

by Yoshitaka Haribara, Shoko Utsunomiya and Yoshihisa Yamamoto

Entropy 2016, 18(4), 151; https://doi.org/10.3390/e18040151 - 19 Apr 2016

Cited by 51 | Viewed by 16793

Abstract

We present the operational principle of a coherent Ising machine (CIM) based on a degenerate optical parametric oscillator (DOPO) network. A quantum theory of CIM is formulated, and the computational ability of CIM is evaluated by numerical simulation based on c-number stochastic differential [...] Read more.

(This article belongs to the Special Issue Quantum Information and Communication: From Foundations to Applications)

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