Principle For Equations Research Articles

Averaging principle for fractional stochastic differential equations driven by Rosenblatt process and Poisson jumps

This manuscript addresses the averaging principle for Riemann-Liouville fractional stochastic differential equations with delays, driven by both the Rosenblatt process and Poisson jumps. Assumptions are made based on the Lipschitz condition, growth condition and averaging condition. Using Cauchy's, Doob's martingale and Gronwall-Bellman inequalities, the solution to the averaged autonomous system is derived, which approximates the solution to the proposed nonautonomous system in the mean square sense. Numerical simulation is provided to understand the obtained results.

Erratum: Moderate Deviations Principle and Central Limit Theorem for Stochastic Cahn-Hilliard Equation in Hölder Norm

We consider a stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise. In this paper, weprove a Central Limit Theorem (CLT) and a Moderate Deviation Principle (MDP) for a perturbed stochastic Cahn-Hilliard equation inHolder norm. The techniques are based on Freidlin-Wentzell’s Large Deviations Principle. The exponential estimates in the space of ̈Holder continuous functions and the Garsia-Rodemich-Rumsey’s lemma plays an important role, an another approach than the Li.R. ̈and Wang.X. Finally, we establish the CLT and MDP for stochastic Cahn-Hilliard equation with uniformly Lipschitzian coefficients. MSC: 60H15, 60F05, 35B40, 35Q62 REFERENCES[1] Ben Arous, G., & Ledoux, M. (1994). Grandes deviations de Freidlin-Wentzell en norme h ́ olderienne. ̈ S ́eminaire de probabilit ́esde Strasbourg, 28, 293-299.[2] Boulanba, L., & Mellouk, M. (2020). Large deviations for a stochastic Cahn–Hilliard equation in Holder norm. ̈ InfiniteDimensional Analysis, Quantum Probability and Related Topics, 23(02), 2050010.[3] Cahn, J. W., & Hilliard, J. E. (1971). Spinodal decomposition: A reprise. Acta Metallurgica, 19(2), 151-161.[4] Cahn, J. W., & Hilliard, J. E. (1958). Free energy of a nonuniform system. I. Interfacial free energy. The Journal of chemicalphysics, 28(2), 258-267.[5] Cardon-Weber, C. (2001). Cahn-Hilliard stochastic equation: existence of the solution and of its density. Bernoulli, 777-816.[6] Chenal, F., & Millet, A. (1997). Uniform large deviations for parabolic SPDEs and applications. Stochastic Processes and theirApplications, 72(2), 161-186.[7] Freidlin, M. I. (1970). On small random perturbations of dynamical systems. Russian Mathematical Surveys, 25(1), 1-55.[8] Li, R., & Wang, X. (2018). Central limit theorem and moderate deviations for a stochastic Cahn-Hilliard equation. arXivpreprint arXiv:1810.05326.[9] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Lecture notes in mathematics, 265-439.[10] Wang, R., & Zhang, T. (2015). Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise.Potential Analysis, 42, 99-113.

Maximum principles for nonlinear integro-differential equations and symmetry of solutions

Maximum principles for nonlinear integro-differential equations and symmetry of solutions

Bank board diversity and environmental risk management: does critical mass matter in project finance decisions?

PurposeThis study investigates the influence of board gender diversity on policy decisions within the banking sector, with a specific focus on the adoption of environmental risk management policies. Drawing on critical mass theory, we provide empirical evidence demonstrating the positive impact of women directors on the implementation of environmental risk management policies aligned with the equator principles (EP) framework.Design/methodology/approachWe analyze data from 540 banks operating in 34 countries over a ten-year period (2013–2022).FindingsOur findings indicate that banks with at least four women directors on their boards are more likely to adopt such policies. Our results remain statistically significant after controlling for endogeneity issues.Practical implicationsThis research advocates for increased female representation on bank boards to foster the adoption of sustainability-focused policies within the banking industry.Originality/valueWe also contribute by providing insights into the role of a diverse board in improving green credit practices among global banks.

The Proof of the Fermar's Last Theorem, Mersenne's Prime Conjecture and Poincare Conjecture in Euclidean Geometry

In order to strictly prove from the point of view of pure mathematics Goldbach's 1742 Goldbach conjecture and Hilbert's twinned prime conjecture in question 8 of his report to the International Congress of Mathematicians in 1900, and the French scholar Alfond de Polignac's 1849 Polignac conjecture, By using Euclid's principle of infinite primes, equivalent transformation principle, and the idea of normalization of set element operation, this paper proves that Goldbach's conjecture, twin primes conjecture and Polignac conjecture are completely correct. In order to strictly prove a conjecture about the solution of positive integers of indefinite equations proposed by French scholar Ferma around 1637 (usually called Ferma's last theorem) from the perspective of pure mathematics, this paper uses the general solution principle of functional equations and the idea of symmetric substitution, as well as the inverse method. It proves that Fermar's last theorem is completely correct.

The twin blow-up method for Hamilton-Jacobi equations in higher dimension

In this paper, we show how to extend the twin blow-up method recently developed by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton-Jacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a {single} blow-up procedure after rescaling the semi-solutions to be compared while two simultaneous blow-ups are performed in this work, one for each variable of the classical doubling variable technique. A one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits plays a key role.

On the obstruction to the Hasse principle for multinorm equations

AbstractWe investigate local-global principles for multinorm equations over a global field. To this extent, we generalize work of Drakokhrust and Platonov to provide explicit and computable formulae for the obstructions to the Hasse principle and weak approximation for multinorm equations. We illustrate the scope of this technique by extending results of Bayer-Fluckiger–Lee–Parimala [1], Demarche–Wei [5] and Pollio [19].

A Large Deviation Principle for Nonlinear Stochastic Wave Equation Driven by Rough Noise

A Large Deviation Principle for Nonlinear Stochastic Wave Equation Driven by Rough Noise

Large deviation principle for backward stochastic differential equations with a stochastic Lipschitz condition on z

In this paper, we provide a probabilistic interpretation of the viscosity solution of a parabolic partial differential equation through the solution of a class of quadratic backward stochastic differential equations (BSDEs). Additionally, we demonstrate the convergence and the large deviation principle for the solutions of these quadratic BSDEs, which are linked to a family of Markov processes with diffusion coefficients that tend toward zero.

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

Does it pay to be Green? The impact of equator principles on project loans

Some financial institutions voluntarily adopt the Equator Principles, requiring them to monitor the environmental and social (ESG) impacts of projects they finance. We investigate the incidence of these costs on corporate borrowers, as well as evidence of benefits in the form of improved loan terms. Using detailed international loan data, we find that borrowers dealing with green banks derive several economic benefits including lower loan spreads. We argue that dealing with ‘green banks’ allows firms to signal their ESG commitment and can help them manage ex post ESG-related risk. The empirical approach also addresses endogeneity concerns. We document other benefits including greater lender participation and the support of Multilateral Development Banks. Our counterfactual analysis, however, shows that if regulatory changes were to encourage working with green banks, firms that only deal with them as a result of the policy would not obtain lower loan spreads.

An invariance principle for the 1D KPZ equation

An invariance principle for the 1D KPZ equation

Large deviation principle of stochastic evolution equations with reflection

In this paper, we establish a large deviation principle for stochastic evolution equations with reflection in an infinite-dimensional ball. Weak convergence approach plays an important role.

Revised and Generalized Results of Averaging Principles for the Fractional Case

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing ε in front of the drift term and ε in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

The Averaging Principle for Caputo Type Fractional Stochastic Differential Equations with Lévy Noise

In this paper, the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise is investigated with consideration of a new method for dealing with singular integrals. Firstly, the estimate on higher moments for the solution is given. Secondly, under some suitable assumptions, we prove the averaging principle for Caputo type fractional stochastic differential equations with Lévy noise by using the Hölder inequality. Finally, a simulation example is given to verify the theoretical results.

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

A singular system involving mixed local and non-local operators

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder’s fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions.

A risk based approach to the principal–agent problem

PurposeWe aim to generalize the continuous-time principal–agent problem to incorporate time-inconsistent utility functions, such as those of mean-variance type, which are prevalent in risk management and finance.Design/methodology/approachWe use recent advancements of the Pontryagin maximum principle for forward-backward stochastic differential equations (FBSDEs) to develop a method for characterizing optimal contracts in such models. This approach addresses the challenges posed by the non-applicability of the classical Hamilton–Jacobi–Bellman equation due to time inconsistency.FindingsWe provide a framework for deriving optimal contracts in the principal–agent problem under hidden action, specifically tailored for time-inconsistent utilities. This is illustrated through a fully solved example in the linear-quadratic setting, demonstrating the practical applicability of the method.Originality/valueThe work contributes to the existing literature by presenting a novel mathematical approach to a class of continuous time principal–agent problems, particularly under hidden action with time-inconsistent utilities, a scenario not previously addressed. The results offer potential insights for both theoretical development and practical applications in finance and economics.

Thermohaline convection in MHD Casson fluid over an exponentially stretching sheet

Abstract This study investigates the thermohaline convection in MHD Casson fluid over an exponentially stretching sheet. This study has practical significance in industrial processes, materials processing, energy systems, and environmental applications. The governing equations describing the conservation for an electrically conducting fluid flow, thermal and concentration transports are considered based on the principles of mass, momentum, energy and concentration equations. Our first step involves transforming the governing nonlinear partial differential equations into a coupled nonlinear ordinary differential equations with the help of suitable similarity transformations. Second step, infinite domain [0, ∞) of the problem to a finite domain [0, 1] through a coordinate transformations. This specific choice is motivated by the wavelet's significance in the finite domain of [0, 1]. Third step, we effectively solve the resulting coupled nonlinear ordinary differential equations using the numerical Hermite wavelet method (HWM). This approach proves to be a valuable technique for obtaining significant results and insights in our study. Finally, the effect of known physical parameters on velocity, temperature and concentration are analysed through tables and graphs.

Predator–prey systems with a variable habitat for predators in advective environments

AbstractCommunity composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction–diffusion–advection predator–prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near‐complete global dynamics of the system. In instances where the functional response to predation conforms to Holling‐type I or II, we explore the uniqueness and stability of positive steady‐state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species.