807 results on '"*STOCHASTIC integrals"'
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2. Dynamic risk measures via backward doubly stochastic Volterra integral equations with jumps.
- Author
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Chen, Yanhong and Miao, Liangliang
- Subjects
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VOLTERRA equations , *STOCHASTIC integrals - Abstract
In this article, we study dynamic risk measures by means of backward doubly stochastic Volterra integral equations (BDSVIEs, for short) with jumps. We establish the well-posedness of BDSVIEs with jumps in the sense of M-solution and prove a comparison theorem of BDSVIEs with jumps. Finally, we study properties of dynamic risk measures induced by BDSVIEs with jumps. Our results extend the well-posedness and the comparison theorem of BDSVIEs without jumps to the setting with jumps, and extend dynamic risk measures induced by BSDEs, BDSDEs, and BSVIEs to the case of BDSVIEs with jumps. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Lagrange interpolation polynomials for solving nonlinear stochastic integral equations.
- Author
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Boukhelkhal, Ikram and Zeghdane, Rebiha
- Subjects
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VOLTERRA equations , *NONLINEAR equations , *NEWTON-Raphson method , *COLLOCATION methods , *STOCHASTIC integrals , *JACOBI polynomials , *NONLINEAR integral equations - Abstract
In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton's methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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4. Generalized location-scale mixtures of elliptical distributions: Definitions and stochastic comparisons.
- Author
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Pu, Tong, Zhang, Yiying, and Yin, Chuancun
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STOCHASTIC integrals , *STOCHASTIC orders , *DISTRIBUTION (Probability theory) , *LINEAR orderings , *ACTUARIAL science - Abstract
This article proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector Z, independent of X and Y, the scale parameter of this class of distributions is mixed with a function α (Z) and its skew parameter is mixed with another function β (Z). Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this class of distributions by means of several stochastic orders including the usual stochastic order, convex order, increasing convex order, supermodular order, and some related linear orders. Two insightful assumptions for the density generators of elliptical distributions, aiming to control the generators' tail, are provided to make stochastic comparisons among mixed-elliptical vectors. Some applications in applied probability and actuarial science are also provided as illustrations on the main findings. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Weighted averaged Gaussian quadrature rules for modified Chebyshev measures.
- Author
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Djukić, Dušan Lj., Mutavdžić Djukić, Rada M., Reichel, Lothar, and Spalević, Miodrag M.
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JACOBI operators , *STOCHASTIC integrals - Abstract
This paper is concerned with the approximation of integrals of a real-valued integrand over the interval [ − 1 , 1 ] by Gauss quadrature. The averaged and optimal averaged quadrature rules ([13,21]) provide a convenient method for approximating the error in the Gauss quadrature. However, they are applicable to all integrands that are continuous on the interval [ − 1 , 1 ] only if their nodes are internal, i.e. if they belong to this interval. We discuss two approaches to determine averaged quadrature rules with nodes in [ − 1 , 1 ] : (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of the first, second, and third kinds that are modified by a linear over linear rational factor, and discuss the internality of averaged, optimal averaged, and truncated optimal averaged quadrature rules. Moreover, we show that the weighting yields internal averaged rules if a weighting parameter is properly chosen, and we provide bounds for this parameter that guarantee internality. Finally, we illustrate that the weighted averaged rules give more accurate estimates of the quadrature error than the truncated optimal averaged rules. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. Connection probabilities of multiple FK-Ising interfaces.
- Author
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Feng, Yu, Peltola, Eveliina, and Wu, Hao
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CONFORMAL field theory , *PROBABILITY theory , *STATISTICAL correlation , *STOCHASTIC integrals , *PARTITION functions - Abstract
We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight q ∈ [ 1 , 4) . Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model ( q = 2 ). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of SLE κ curves (with κ = 16 / 3 for q = 2 ). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all q ∈ [ 1 , 4) , thus providing further evidence of the expected CFT description of these models. [ABSTRACT FROM AUTHOR]
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- 2024
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7. Stochastic Volterra equations with time-changed Lévy noise and maximum principles.
- Author
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di Nunno, Giulia and Giordano, Michele
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VOLTERRA equations , *STOCHASTIC differential equations , *NOISE , *NATURAL resources , *STOCHASTIC integrals - Abstract
Motivated by a problem of optimal harvesting of natural resources, we study a control problem for Volterra type dynamics driven by time-changed Lévy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative introduced in Di Nunno and Eide (Stoch Anal Appl 28:54-85, 2009). We prove both a sufficient and necessary stochastic maximum principle. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. A representation theorem for set-valued submartingales.
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T. Tuyen, Luc and T. Luan, Vu
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PROBABILITY theory , *STOCHASTIC integrals , *INTEGRAL representations , *MARTINGALES (Mathematics) , *RANDOM sets , *STOCHASTIC processes - Abstract
The integral representation theorem for martingales has been widely used in probability theory. In this work, we propose and prove a general representation theorem for a class of set-valued submartingales. We also extend the stochastic integral representation for non-trivial initial set-valued martingales. Moreover, we show that this result covers the existing ones in the literature for both degenerated and non-degenerated set-valued martingales. [ABSTRACT FROM AUTHOR]
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- 2024
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9. Functional Solutions of Stochastic Differential Equations.
- Author
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van den Berg, Imme
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STOCHASTIC differential equations , *STOCHASTIC integrals , *DIFFERENTIAL equations , *FUNCTIONAL differential equations , *PARTIAL differential equations , *ORDINARY differential equations - Abstract
We present an integration condition ensuring that a stochastic differential equation d X t = μ (t , X t) d t + σ (t , X t) d B t , where μ and σ are sufficiently regular, has a solution of the form X t = Z (t , B t) . By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form X t = Z (t , Y t) , with Y t an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green's Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point (t , x) , we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. A projection method based on the piecewise Chebyshev cardinal functions for nonlinear stochastic ABC fractional integro‐differential equations.
- Author
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Heydari, M. H., Zhagharian, Sh., and Cattani, C.
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NONLINEAR functions , *STOCHASTIC integrals , *ALGEBRAIC equations , *INTEGRO-differential equations , *CAPUTO fractional derivatives , *MATRICES (Mathematics) , *FRACTIONAL differential equations - Abstract
In this study, the Atangana–Baleanu fractional derivative in the Caputo type (as a kind of non‐local and non‐singular derivative) is used to define a new class of stochastic fractional integro‐differential equations. A projection method (more precisely, a Galerkin approach) based on the piecewise Chebyshev cardinal functions is developed to solve these stochastic fractional equations. To construct this method, the operational matrices of fractional and stochastic integrals of these basis functions are obtained and used in the established method. By approximating the solution of the problem with a finite expansion of the expressed basis functions (in which the expansion coefficients are unknown), a system of algebraic equations is obtained. By solving this system, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are obtained. The convergence analysis of the proposed method is investigated, theoretically and numerically. The accuracy of the established procedure is illustrated by solving several numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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11. Nearly unstable integer‐valued ARCH process and unit root testing.
- Author
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Barreto‐Souza, Wagner and Chan, Ngai Hang
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ASYMPTOTIC distribution , *TIME series analysis , *LIMIT theorems , *DEATH rate , *STOCHASTIC integrals , *TOPOLOGY , *ERROR rates , *MONTE Carlo method - Abstract
This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom. [ABSTRACT FROM AUTHOR]
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- 2024
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12. Backward doubly stochastic differential equations driven by fractional Brownian motion with stochastic integral-Lipschitz coefficients.
- Author
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Ndiaye, Assane, Aidara, Sadibou, and Sow, Ahmadou Bamba
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FRACTIONAL differential equations , *STOCHASTIC differential equations , *BROWNIAN motion , *STOCHASTIC integrals - Abstract
This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1 2 . We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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13. BSDEs driven by fractional Brownian motion with time-delayed generators.
- Author
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Aidara, Sadibou and Sylla, Lamine
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BROWNIAN motion , *STOCHASTIC differential equations , *STOCHASTIC integrals , *FRACTIONAL differential equations , *MOVING average process , *TIME perspective - Abstract
This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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14. Lagrangian stochastic integrals of motion in isotropic random flows.
- Author
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Sirota, V. A., Il'yn, A. S., Kopyev, A. V., and Zybin, K. P.
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STOCHASTIC integrals , *INTEGRALS - Abstract
A set of exact integrals of motion is found for systems driven by homogenous isotropic stochastic flow. The integrals of motion describe the evolution of (hyper-)surfaces of different dimensions transported by the flow and can be expressed in terms of local surface densities. The expression for the integrals is universal: it represents general geometric properties and does not depend on the statistics of the specific flow. [ABSTRACT FROM AUTHOR]
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- 2024
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15. Accuracy of approximate methods for the calculation of absorption-type linear spectra with a complex system–bath coupling.
- Author
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Nöthling, J. A., Mančal, Tomáš, and Krüger, T. P. J.
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LINEAR dichroism , *PATH integrals , *STOCHASTIC integrals , *BAND gaps , *CIRCULAR dichroism , *CHLOROPHYLL spectra , *DIMERS - Abstract
The accuracy of approximate methods for calculating linear optical spectra depends on many variables. In this study, we fix most of these parameters to typical values found in photosynthetic light-harvesting complexes of plants and determine the accuracy of approximate spectra with respect to exact calculation as a function of the energy gap and interpigment coupling in a pigment dimer. We use a spectral density with the first eight intramolecular modes of chlorophyll a and include inhomogeneous disorder for the calculation of spectra. We compare the accuracy of absorption, linear dichroism, and circular dichroism spectra calculated using the Full Cumulant Expansion (FCE), coherent time-dependent Redfield (ctR), and time-independent Redfield and modified Redfield methods. As a reference, we use spectra calculated with the exact stochastic path integral evaluation method. We find the FCE method to be the most accurate for the calculation of all spectra. The ctR method performs well for the qualitative calculation of absorption and linear dichroism spectra when the pigments are moderately coupled (∼ 15 c m − 1 ) , but ctR spectra may differ significantly from exact spectra when strong interpigment coupling (> 100 c m − 1 ) is present. The dependence of the quality of Redfield and modified Redfield spectra on molecular parameters is similar, and these methods almost always perform worse than ctR, especially when the interpigment coupling is strong or the excitonic energy gap is small (for a given coupling). The accuracy of approximate spectra is not affected by resonance with intramolecular modes for typical system–bath coupling and disorder values found in plant light-harvesting complexes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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16. Adaptive state feedback control of output‐constrained stochastic nonlinear systems with stochastic integral input‐to‐state stability inverse dynamics.
- Author
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Xie, Ruiming and Xu, Shengyuan
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STATE feedback (Feedback control systems) , *STOCHASTIC systems , *ADAPTIVE fuzzy control , *STOCHASTIC integrals , *CLOSED loop systems , *ADAPTIVE control systems , *NONLINEAR systems , *NONLINEAR functions - Abstract
This article studies the adaptive state‐feedback control problem of output‐constrained stochastic high‐order nonlinear systems with stochastic integral input‐to‐state stability (SiISS) inverse dynamics. A key nonlinear transformation function is constructed to convert the original output‐constrained stochastic nonlinear system into an equivalent form without any output constraint. By subtly using the SiISS small‐gain condition and fully extracting the characteristics of system nonlinearities, two new control design and analysis methods are developed to guarantee that the closed‐loop system has an almost surely unique solution, all the closed‐loop signals are bounded almost surely, and the equilibrium point is stable in probability without the violation of output constraint. A simulation result is provided to show the effectiveness of this control method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. Stochastic calculus for tempered fractional Brownian motion and stability for SDEs driven by TFBM.
- Author
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Zhang, Lijuan, Wang, Yejuan, and Hu, Yaozhong
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BROWNIAN motion , *STOCHASTIC integrals , *STOCHASTIC differential equations , *FRACTIONAL calculus , *MALLIAVIN calculus - Abstract
The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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18. Jacobi polynomials for the numerical solution of multi-dimensional stochastic multi-order time fractional diffusion-wave equations.
- Author
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Heydari, M.H., Zhagharian, Sh., and Razzaghi, M.
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JACOBI polynomials , *ALGEBRAIC equations , *STOCHASTIC integrals , *MATRICES (Mathematics) , *EQUATIONS , *HAMILTON-Jacobi equations - Abstract
In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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19. On the stochastic differentiability of noncausal processes with respect to the process with quadratic variation.
- Author
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Hoshino, Kiyoiki
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STOCHASTIC integrals , *STOCHASTIC processes , *PROBLEM solving , *PROBABILITY theory , *QUADRATIC forms - Abstract
Let (V t) t ∈ [ 0 , L ] be a stochastic process with quadratic variation on a probability space (Ω , F , P) and Q (∋ 0) a dense subset of [ 0 , L ] , where [ 0 , L ] is regarded as the infinite interval [ 0 , ∞) when L = ∞. First, we introduce the L 0 (Ω) -module D Q (V) of V-differentiable noncausal processes on Q and V-derivative operator D V , Q = d Q d Q V defined on D Q (V) , which enjoys the modularity: D V , Q (α X + β Y) = α D V , Q X + β D V , Q Y for any X , Y ∈ D Q (V) and α , β ∈ L 0 (Ω). Second, we show that the class Q V , Q = { X ∈ D Q (V) | d [ X ] Q , t d [ V ] Q , t = | d Q X t d Q V t | 2 } forms an L 0 (Ω) -module, where [ ] Q , t stands for the quadratic variation on Q. As a result, we have the isometry: 〈 X , Y 〉 Q , t = 〈 D V , Q X , D V , Q Y 〉 L 2 ([ 0 , t ] , [ V ]) for any X , Y ∈ Q V , Q , where 〈 , 〉 Q , t stands for the quadratic covariation on Q. Finally, we present universal properties and examples of the stochastic integral I with D V , Q ∘ I = i d D (I) . This result is essentially used for solving the identification problem from the stochastic Fourier coefficients. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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20. Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures.
- Author
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Soize, Christian
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PROBABILITY measures , *FOURIER transforms , *MACHINE learning , *STOCHASTIC models , *INVERSE problems , *STOCHASTIC integrals , *PROBABILITY theory - Abstract
This paper deals with the taking into account a given target set of realizations as constraints in the Kullback–Leibler divergence minimum principle (KLDMP). We present a novel probabilistic learning algorithm that makes it possible to use the KLDMP when the constraints are not defined by a target set of statistical moments for the quantity of interest (QoI) of an uncertain/stochastic computational model, but are defined by a target set of realizations for the QoI for which the statistical moments associated with these realizations are not or cannot be estimated. The method consists in defining a functional constraint, as the equality of the Fourier transforms of the posterior probability measure and the target probability measure, and in constructing a finite representation of the weak formulation of this functional constraint. The proposed approach allows for estimating the posterior probability measure of the QoI (unsupervised case) or of the posterior joint probability measure of the QoI with the control parameter (supervised case). The existence and the uniqueness of the posterior probability measure is analyzed for the two cases. The numerical aspects are detailed in order to facilitate the implementation of the proposed method. The presented application in high dimension demonstrates the efficiency and the robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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21. Propagation of chaos for maxima of particle systems with mean-field drift interaction.
- Author
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Kolliopoulos, Nikolaos, Larsson, Martin, and Zhang, Zeyu
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STOCHASTIC analysis , *ITERATED integrals , *STOCHASTIC integrals , *ARGUMENT - Abstract
We study the asymptotic behavior of the normalized maxima of real-valued diffusive particles with mean-field drift interaction. Our main result establishes propagation of chaos: in the large population limit, the normalized maxima behave as those arising in an i.i.d. system where each particle follows the associated McKean–Vlasov limiting dynamics. Because the maximum depends on all particles, our result does not follow from classical propagation of chaos, where convergence to an i.i.d. limit holds for any fixed number of particles but not all particles simultaneously. The proof uses a change of measure argument that depends on a delicate combinatorial analysis of the iterated stochastic integrals appearing in the chaos expansion of the Radon–Nikodym density. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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22. On Ulam type of stability for stochastic integral equations with Volterra noise.
- Author
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Bishop, Sheila A. and Iyase, Samuel A.
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VOLTERRA equations , *STOCHASTIC processes , *STOCHASTIC integrals , *NOISE - Abstract
This paper concerns the existence, uniqueness and stability of solutions of stochastic Volterra integral equations perturbed by some random processes. The obtained results extend, generalize and enrich the theory of stochastic Volterra integral equations in literature. Lastly, for illustration, we give an example that agrees with the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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23. The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem.
- Author
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Blackstone, Elliot, Charlier, Christophe, and Lenells, Jonatan
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POINT processes , *STOCHASTIC integrals , *RIEMANN surfaces , *EIGENVALUES , *TORUS - Abstract
The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability P∃no points in the Bessel process on(0,x1)∪(x2,x3)∪⋯∪(x2g,x2g+1),$$\begin{equation*} {\mathbb{P}\left(\exists \text{ no points in the Bessel process on}\ (0,{x}_{1})\cup ({x}_{2},{x}_{3})\cup \cdots \cup ({x}_{2g},{x}_{2g+1})\right),} \end{equation*}$$where 0
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- 2023
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24. Existence and Uniqueness Theorem of Fuzzy Stochastic Ordinary Differential Equations.
- Author
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Kareem, Nabaa R., Fadhel, Fadhel S., and Al-Nassir, Sadiq
- Subjects
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ORDINARY differential equations , *EXISTENCE theorems , *STOCHASTIC integrals , *MATHEMATICAL induction , *BROWNIAN motion , *STOCHASTIC differential equations - Abstract
A fuzzy valued diffusion term, which in a fuzzy stochastic differential equation refers to one-dimensional Brownian motion, is defined by the meaning of the stochastic integral of a fuzzy process. In this paper, the existence and uniqueness theorem of fuzzy stochastic ordinary differential equations, based on the mean square convergence of the mathematical induction approximations to the associated stochastic integral equation, are stated and demonstrated. [ABSTRACT FROM AUTHOR]
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- 2023
- Full Text
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25. SPIKE VARIATIONS FOR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS.
- Author
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TIANXIAO WANG and JIONGMIN YONG
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VOLTERRA equations , *STOCHASTIC integrals , *PONTRYAGIN'S minimum principle , *STOCHASTIC control theory , *STOCHASTIC differential equations , *QUADRATIC forms , *MAXIMUM principles (Mathematics) - Abstract
The spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use It's formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well. [ABSTRACT FROM AUTHOR]
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- 2023
- Full Text
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26. MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF STOCHASTIC EVOLUTION EQUATIONS WITH RECURSIVE UTILITIES.
- Author
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GUOMIN LIU and SHANJIAN TANG
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STOCHASTIC control theory , *MAXIMUM principles (Mathematics) , *EVOLUTION equations , *STOCHASTIC differential equations , *STOCHASTIC integrals , *INTEGRAL equations - Abstract
We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain to not be convex and the generator of the BSDE to vary with the second unknown variable z. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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27. Stochastic inclusions and set-valued stochastic equations with mixed integrals in the plane.
- Author
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Michta, Mariusz
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INTEGRAL equations , *TOPOLOGICAL property , *MARTINGALES (Mathematics) , *STOCHASTIC integrals , *DIFFERENTIAL inclusions - Abstract
The paper refers to the study of properties of solutions to two-parameter stochastic inclusions and set-valued stochastic equations with set-valued mixed integrals driven by finite variation processes and martingales. We present new types of such inclusions and equations that generalize those studied earlier. Apart from existence results to such inclusions and equations also topological properties of their solutions are studied. Additionally some connections between their solutions are established. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in deterministic and stochastic cases. [ABSTRACT FROM AUTHOR]
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- 2023
- Full Text
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28. Andriy Anatoliyovych Dorogovtsev (On His 60th Birthday).
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STOCHASTIC analysis , *STOCHASTIC integrals , *GAUSSIAN processes , *STOCHASTIC differential equations , *STOCHASTIC processes - Abstract
Andriy Anatoliyovych Dorogovtsev, a prominent Ukrainian mathematician, recently celebrated his 60th birthday. He is a Corresponding Member of the National Academy of Sciences of Ukraine, a doctor of physical and mathematical sciences, a professor, and the Head of the Department of the Theory of Random Processes at the Institute of Mathematics of the National Academy of Sciences of Ukraine. Dorogovtsev has made significant contributions to the field of mathematics, particularly in the areas of stochastic analysis, stochastic flows, and measure-valued processes. He is also known for his dedication to the popularization of mathematics and the training of young scientists. Dorogovtsev's colleagues and disciples admire his professionalism, passion for mathematics, and integrity in all aspects of life. [Extracted from the article]
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- 2024
- Full Text
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29. The heat kernel of the asymmetric quantum Rabi model.
- Author
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Reyes-Bustos, Cid
- Subjects
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RABI oscillations , *STOCHASTIC integrals , *PARTITION functions , *ZETA functions , *PATH integrals , *SYMMETRY breaking - Abstract
In this paper we derive an explicit formula for the heat kernel of the asymmetric quantum Rabi model, a symmetry breaking generalization of the quantum Rabi model (QRM). The method described here is the extension of a recently developed method for the heat kernel of the QRM that uses the Trotter–Kato product formula instead of path integrals or stochastic methods. In addition to the heat kernel formula, we give applications including the explicit formula for the partition function and the Weyl law for the distribution of the eigenvalues, obtained from the corresponding spectral zeta function. [ABSTRACT FROM AUTHOR]
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- 2023
- Full Text
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30. Central limit type theorem and large deviation principle for multi-scale McKean–Vlasov SDEs.
- Author
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Hong, Wei, Li, Shihu, Liu, Wei, and Sun, Xiaobin
- Subjects
- *
CENTRAL limit theorem , *LARGE deviations (Mathematics) , *STOCHASTIC differential equations , *STOCHASTIC systems , *STOCHASTIC integrals , *DYNAMICAL systems - Abstract
The main aim of this work is to study the asymptotic behavior for multi-scale McKean–Vlasov stochastic dynamical systems. Firstly, we obtain a central limit type theorem, i.e. the deviation between the slow component X ε and the solution X ¯ of the averaged equation converges weakly to a limiting process. More precisely, X ε - X ¯ ε converges weakly in C ([ 0 , T ] , R n) to the solution of certain distribution dependent stochastic differential equation, which involves an extra explicit stochastic integral term. Secondly, in order to estimate the probability of deviations away from the limiting process, we further investigate the Freidlin–Wentzell's large deviation principle for multi-scale McKean–Vlasov stochastic system when the small-noise regime parameter δ → 0 and the time scale parameter ε (δ) satisfies ε (δ) / δ → 0 . The main techniques are based on the Poisson equation for central limit type theorem and the weak convergence approach for large deviation principle. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. A Strong Averaging Principle Rate for Two-Time-Scale Coupled Forward–Backward Stochastic Differential Equations Driven by Fractional Brownian Motion.
- Author
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Xu, Jie, Lian, Qiqi, and Wu, Jiang-Lun
- Subjects
- *
BROWNIAN motion , *STOCHASTIC differential equations , *FRACTIONAL differential equations , *MALLIAVIN calculus , *STOCHASTIC integrals , *DISCRETIZATION methods - Abstract
This paper concerns the strong convergence rate of an averaging principle for two-time-scale coupled forward–backward stochastic differential equations (CFBSDEs, for short) driven by fractional Brownian motion (fBm, for short). The fast component is a forward stochastic differential equation (FSDE, for short) driven by Brownian motion, while the slow component is a backward stochastic differential equation (BSDE, for short) driven by fBm with the Hurst index greater than 1/2. Combining Malliavin calculus theory with stochastic integral and Khasminskii's time discretization method, the rate of strong convergence for the slow component towards the solution of the averaging equation in the mean square sense is derived. The strong convergence rate of an averaging principle for fast–slow CFBSDEs driven by fBm is new. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. Inference for the VEC(1) model with a heavy-tailed linear process errors.
- Author
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Guo, Feifei and Ling, Shiqing
- Subjects
- *
STOCHASTIC integrals , *STOCHASTIC processes , *LEAST squares , *FLOUR , *SAMPLING (Process) - Abstract
This article studies the first-order vector error correction (VEC(1)) model when its noise is a linear process of independent and identically distributed (i.i.d.) heavy-tailed random vectors with a tail index α ∈ (0 , 2) . We show that the rate of convergence of the least squares estimator (LSE) related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes, while the LSE related to the short-term parameters is not consistent. We further propose an automated approach via adaptive shrinkage techniques to determine the cointegrating rank in the VEC(1) model. It is demonstrated that the cointegration rank r0 can be consistently selected despite the fact that the LSE related to the short-term parameters is not consistently estimable when the tail index α ∈ (1 , 2) . Simulation studies are carried out to evaluate the performance of the proposed procedure in finite samples. Last, we use our techniques to explore the long-run and short-run behavior of the monthly prices of wheat, corn, and wheat flour in the United States. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. The Existence and Averaging Principle for a Class of Fractional Hadamard Itô–Doob Stochastic Integral Equations.
- Author
-
Rhaima, Mohamed, Mchiri, Lassaad, and Ben Makhlouf, Abdellatif
- Subjects
- *
STOCHASTIC integrals , *GRONWALL inequalities , *FRACTIONAL integrals , *STOCHASTIC systems - Abstract
In this paper, we investigate the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also unveils the remarkable averaging principle within FHIDSIE. To accomplish this, we harness powerful mathematical tools, including the Hölder and Gronwall inequalities. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. On Traces of Linear Operators with Symmetrized Volterra-Type Kernels.
- Author
-
Rybakov, Konstantin
- Subjects
- *
STOCHASTIC integrals , *STOCHASTIC differential equations , *ORTHOGONAL functions - Abstract
A solution to the trace convergence problem, which arises in proving the mean-square convergence for the approximation of iterated Stratonovich stochastic integrals, is proposed. This approximation is based on the representation of factorized Volterra-type functions as the orthogonal series. Solving the trace convergence problem involves the theory of trace class operators for symmetrized Volterra-type kernels. The main results are primarily focused on the approximation of iterated Stratonovich stochastic integrals, which are used to implement numerical methods for solving stochastic differential equations based on the Taylor–Stratonovich expansion. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. NEW INEQUALITIES OF HERMITE–HADAMARD TYPE FOR n-POLYNOMIAL s-TYPE CONVEX STOCHASTIC PROCESSES.
- Author
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KALSOOM, HUMAIRA and KHAN, ZAREEN A.
- Subjects
- *
STOCHASTIC processes , *INTEGRAL inequalities , *STOCHASTIC analysis , *REAL numbers , *CONVEX functions , *INTEGRAL functions , *STOCHASTIC integrals - Abstract
The purpose of this paper is to introduce a more generalized class of convex stochastic processes and explore some of their algebraic properties. This new class of stochastic processes is called the n -polynomial s -type convex stochastic process. We demonstrate that this new class of stochastic processes leads to the discovery of novel Hermite–Hadamard type inequalities. These inequalities provide upper bounds on the integral of a convex function over an interval in terms of the moments of the stochastic process and the convexity parameter s. To compare the effectiveness of the newly discovered Hermite–Hadamard type inequalities, we also consider other commonly used integral inequalities, such as Hölder, Hölder–Ïşcan, and power-mean, as well as improved power-mean integral inequalities. We show that the Hölder–Ïşcan and improved power-mean integral inequalities provide a better approach for the n -polynomial s -type convex stochastic process than the other integral inequalities. Finally, we provide some applications of the Hermite–Hadamard type inequalities to special means of real numbers. Our findings provide a useful tool for the analysis of stochastic processes in various fields, including finance, economics, and engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. The stochastic Leibniz formula for Volterra integrals under enlarged filtrations.
- Author
-
Hess, Markus
- Subjects
- *
VOLTERRA equations , *STOCHASTIC integrals , *INTEGRALS , *BOND prices , *STOCHASTIC differential equations - Abstract
In this paper, we derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these setups, we compare the ordinary with the stochastic (Doleans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type and infer the associated bond price representation. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics.
- Author
-
Rybakov, Konstantin
- Subjects
- *
STOCHASTIC integrals , *ITERATED integrals , *STOCHASTIC models , *STOCHASTIC differential equations , *MALLIAVIN calculus - Abstract
Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multiplicity, including integrals from Taylor–Itô and Taylor–Stratonovich expansions, are obtained by the spectral method. They are required for the implementation of numerical methods for solving Itô and Stratonovich stochastic differential equations with high orders of mean-square and strong convergence. The purpose of such numerical methods is the modeling of nonlinear stochastic dynamics in many fields. This paper contains necessary theoretical results, as well as the results of numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
38. A Stochastic Maximum Principle for Partially Observed Optimal Control Problem of Mckean–Vlasov FBSDEs with Random Jumps.
- Author
-
Abba, Khedidja and Lakhdari, Imad Eddine
- Subjects
- *
STOCHASTIC differential equations , *MAXIMUM principles (Mathematics) , *RANDOM measures , *BROWNIAN motion , *STOCHASTIC integrals , *PROBABILITY measures - Abstract
In this paper, we study the stochastic maximum principle for partially observed optimal control problem of forward–backward stochastic differential equations of McKean–Vlasov type driven by a Poisson random measure and an independent Brownian motion. The coefficients of the system and the cost functional depend on the state of the solution process as well as of its probability law and the control variable. Necessary and sufficient conditions of optimality for this systems are established under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov's theorem and the derivatives with respect to probability law. As an illustration, a partially observed linear–quadratic control problem of McKean–Vlasov forward–backward stochastic differential equations type is studied in terms of the stochastic filtering. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. A novel numerical approach based on shifted second‐kind Chebyshev polynomials for solving stochastic Itô–Volterra integral equation of Abel type with weakly singular kernel.
- Author
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Saha Ray, Santanu and Gupta, Reema
- Subjects
- *
VOLTERRA equations , *CHEBYSHEV polynomials , *STOCHASTIC integrals , *FREDHOLM equations , *SINGULAR integrals , *ALGEBRAIC equations , *MATRICES (Mathematics) , *COLLOCATION methods - Abstract
In this paper, a collocation method based on shifted second‐order Chebyshev polynomials is implemented to obtain the approximate solution of the stochastic Itô–Volterra integral equation of Abel type with weakly singular kernel. In this method, operational matrices are used to convert the stochastic Itô–Volterra integral equation to algebraic equations that are linear. The algorithm of the proposed numerical scheme has been presented in this paper. Also, the error bound and convergence of the proposed method are well established. Consequently, two illustrative examples are provided to demonstrate the efficiency, plausibility, reliability, and consistency of the current methodology. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
40. Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation.
- Author
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Pripoae, Cristina-Liliana, Hirica, Iulia-Elena, Pripoae, Gabriel-Teodor, and Preda, Vasile
- Subjects
- *
GEOMETRIC modeling , *TENSOR fields , *STOCHASTIC integrals , *EQUATIONS , *INTEGRAL equations , *HELMHOLTZ equation , *GIBBS sampling - Abstract
By replacing the internal energy with the free energy, as coordinates in a "space of observables", we slightly modify (the known three) non-holonomic geometrizations from Udriste's et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R 4 as the "graph" of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general "equivalence problem" between classical entropy and statistical entropy. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations.
- Author
-
Samar, Mahvish, Kutorzi, Edwin Yao, and Zhu, Xinzhong
- Subjects
- *
VOLTERRA equations , *STOCHASTIC integrals , *MATRICES (Mathematics) , *NEWTON-Raphson method , *STOCHASTIC matrices - Abstract
This work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton's method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operational matrix of integration. We present examples to illustrate the estimate analysis and to demonstrate the convergence of the two approximating sequences separately. To measure their accuracy, we compare the solutions with values of exact and approximative solutions at a few selected locations using a specified absolute error. We also propose an efficient method for solving a triangular linear algebraic problem using a single integral equation. To confirm the effectiveness of our method, we conduct numerical experiments with issues from real-world applications. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
42. Accurate and stable numerical method based on the Floater-Hormann interpolation for stochastic Itô-Volterra integral equations.
- Author
-
Mirzaee, Farshid, Naserifar, Shiva, and Solhi, Erfan
- Subjects
- *
VOLTERRA equations , *STOCHASTIC integrals , *INTEGRAL equations , *ALGEBRAIC equations , *MATHEMATICAL physics - Abstract
In various fields of science and engineering, such as financial mathematics, mathematical physics models, and radiation transfer, stochastic integral equations are important and practical tools for modeling and describing problems. Due to the existence of random factors, we face a fundamental problem in solving stochastic integral equations, and that is the lack of analytical solutions or the great complexity of these solutions. Therefore, finding an efficient numerical solution is essential. In this paper, we intend to propose and study a new method based on the Floater-Hormann interpolation and the spectral collocation method for linear and nonlinear stochastic Itô-Volterra integral equations (SVIEs). The Floater-Hormann interpolation offers an approximation regardless of the distribution of the points. Therefore, this method can be mentioned as a meshless method. The presented method reduces SVIEs under consideration into a system of algebraic equations that can be solved by the appropriate method. We presented an error bound to be sure of the convergence and reliability of the method. Finally, the efficiency and the applicability of the present scheme are investigated through some numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Hedging portfolio for a market model of degenerate diffusions.
- Author
-
Çağlar, Mine, Demirel, İhsan, and Üstünel, Ali Süleyman
- Subjects
- *
MALLIAVIN calculus , *STOCHASTIC integrals , *HEDGING (Finance) , *LINEAR equations , *TIME perspective - Abstract
We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark–Hausmann–Bismut–Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation at the first place. We demonstrate the hedging strategy as explicitly as possible with some examples of the payoff function such as those used in exotic options, whose value at maturity depends on the prices over the entire time horizon. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
44. CBI-time-changed Lévy processes.
- Author
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Fontana, Claudio, Gnoatto, Alessandro, and Szulda, Guillaume
- Subjects
- *
LEVY processes , *BRANCHING processes , *STOCHASTIC integrals , *STOCHASTIC processes , *INTEGRAL equations - Abstract
We introduce and study the class of CBI-time-changed Lévy processes (CBITCL), obtained by time-changing a Lévy process with respect to an integrated continuous-state branching process with immigration (CBI). We characterize CBITCL processes as solutions to a certain stochastic integral equation and relate them to affine stochastic volatility processes. We provide a complete analysis of the time of explosion of exponential moments of CBITCL processes and study their asymptotic behavior. In addition, we show that CBITCL processes are stable with respect to a suitable class of equivalent changes of measure. As illustrated by some examples, CBITCL processes are flexible and tractable processes with a significant potential for applications in finance. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
45. Delay BSDEs driven by fractional Brownian motion.
- Author
-
Aidara, Sadibou and Sane, Ibrahima
- Subjects
- *
BROWNIAN motion , *STOCHASTIC differential equations , *STOCHASTIC integrals , *FRACTIONAL differential equations - Abstract
This paper deals with a class of delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 ). In this type of equation, a generator at time t can depend not only on the present but also on the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is a divergence-type integral. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
46. A Schrödinger random operator with semimartingale potential.
- Author
-
Gutierrez-Pavón, Jonathan J. and Pacheco, Carlos G.
- Subjects
- *
RANDOM operators , *SCHRODINGER operator , *STOCHASTIC integrals , *MARTINGALES (Mathematics) , *BILINEAR forms , *SEMILINEAR elliptic equations - Abstract
We study a Schrödinger random operator where the potential is in terms of a continuous semimartingale. Our model is a generalization of the well-known case where the potential is the white-noise. Our approach is to analyze the random operator by means of its bilinear form. This allows us to construct an inverse operator using an explicit Green kernel. To characterize such homogeneous solutions we use certain stochastic equations in terms of stochastic integrals with respect to the semimartingale. An important tool that we use is the multi-dimensional Itô formula. Also, one important corollary of our results is that the operator has a discrete spectrum. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. On the Ayed-Kuo stochastic integration for anticipating integrands.
- Author
-
Jornet, Marc
- Subjects
- *
STOCHASTIC integrals , *INTEGRALS - Abstract
In this article, we prove new results for the anticipating stochastic integral introduced by Ayed and Kuo. We present an existence criterion for the integral, a Fubini's theorem, a Leibniz integral rule, and an alternative definition for a class of integrands. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
48. Hermite-Hadamard Inequalities Type Using Fractional Integrals for MT-convex Stochastic Process.
- Author
-
Rholam, O., Barmaki, M., and Gretete, D.
- Subjects
- *
STOCHASTIC integrals , *STOCHASTIC processes , *FRACTIONAL integrals , *ABSOLUTE value , *INTEGRAL operators - Abstract
By applying the standard fractional integral operator of Riemann-Liouville onMT-convex stochastic processes, we can obtain new inequalities of Hermite-Hadamard, providing in the process new estimates on these types of Hermite-Hadamard inequalities for stochastic process whose first derivatives absolute values are MT-convex. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
49. Analysis of a stochastic SVIR model with time‐delayed stages of vaccination and Lévy jumps.
- Author
-
Mehdaoui, Mohamed, Alaoui, Abdesslem Lamrani, and Tilioua, Mouhcine
- Subjects
- *
STOCHASTIC analysis , *STOCHASTIC models , *STOCHASTIC differential equations , *DELAY differential equations , *MATERNALLY acquired immunity , *ANTI-vaccination movement , *STOCHASTIC integrals - Abstract
The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long‐period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible‐Vaccinated‐Infected‐Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy‐type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well‐posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping‐time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. The Itô integral and near-martingales in Riesz spaces.
- Author
-
Divandar, Mahin Sadat and Sadeghi, Ghadir
- Subjects
- *
RIESZ spaces , *STOCHASTIC integrals , *BROWNIAN motion , *INTEGRALS , *STOCHASTIC processes - Abstract
A new class of the Itô integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is a motivitation to construct near-martingales in Riesz spaces. Furthermore, we state Doob–Meyer decomposition theorem for near-submartingales in Riesz spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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