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Journal articles on the topic 'Stochastic Differential Algebraic Equations'

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Author: Grafiati

Published: 6 September 2023

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1

Alabert, Aureli, and Marco Ferrante. "Linear stochastic differential-algebraic equations with constant coefficients." Electronic Communications in Probability 11 (2006): 316–35. http://dx.doi.org/10.1214/ecp.v11-1236.

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2

Higueras, I., J. Moler, F. Plo, and M. San Miguel. "Urn models and differential algebraic equations." Journal of Applied Probability 40, no. 2 (June 2003): 401–12. http://dx.doi.org/10.1239/jap/1053003552.

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The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

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3

Higueras, I., J. Moler, F. Plo, and M. San Miguel. "Urn models and differential algebraic equations." Journal of Applied Probability 40, no. 02 (June 2003): 401–12. http://dx.doi.org/10.1017/s0021900200019380.

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The aim of this paper is to study the distribution of colours, { X n }, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process { X n } is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process { X n }.

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4

Pulch, Roland. "Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations." Journal of Computational and Applied Mathematics 262 (May 2014): 281–91. http://dx.doi.org/10.1016/j.cam.2013.10.046.

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5

Li, Xun, Jingtao Shi, and Jiongmin Yong. "Mean-field linear-quadratic stochastic differential games in an infinite horizon." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 81. http://dx.doi.org/10.1051/cocv/2021078.

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This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

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6

CONG, NGUYEN DINH, and NGUYEN THI THE. "LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1." Stochastics and Dynamics 12, no. 04 (October 10, 2012): 1250002. http://dx.doi.org/10.1142/s0219493712500025.

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We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

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7

Lv, Xueqin, and Jianfang Gao. "Treatment for third-order nonlinear differential equations based on the Adomian decomposition method." LMS Journal of Computation and Mathematics 20, no. 1 (2017): 1–10. http://dx.doi.org/10.1112/s1461157017000018.

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The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

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8

Drăgan, Vasile, Ivan Ganchev Ivanov, and Ioan-Lucian Popa. "A Game — Theoretic Model for a Stochastic Linear Quadratic Tracking Problem." Axioms 12, no. 1 (January 11, 2023): 76. http://dx.doi.org/10.3390/axioms12010076.

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In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations.

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9

Curry, Charles, Kurusch Ebrahimi–Fard, Simon J. A. Malham, and Anke Wiese. "Algebraic structures and stochastic differential equations driven by Lévy processes." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2221 (January 2019): 20180567. http://dx.doi.org/10.1098/rspa.2018.0567.

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We construct an efficient integrator for stochastic differential systems driven by Lévy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover, the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Lévy processes.

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10

Nair, Priya, and Anandaraman Rathinasamy. "Stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential algebraic equations." Results in Applied Mathematics 12 (November 2021): 100187. http://dx.doi.org/10.1016/j.rinam.2021.100187.

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11

Saleh, M. M., I. L. El-Kalla, and M. M. Ehab. "Stochastic Finite Element Technique for Stochastic One-Dimension Time-Dependent Differential Equations with Random Coefficients." Differential Equations and Nonlinear Mechanics 2007 (2007): 1–16. http://dx.doi.org/10.1155/2007/48527.

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The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.

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12

Zhou, Haiying, Huainian Zhu, and Chengke Zhang. "Linear Quadratic Nash Differential Games of Stochastic Singular Systems." Journal of Systems Science and Information 2, no. 6 (December 25, 2014): 553–60. http://dx.doi.org/10.1515/jssi-2014-0553.

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AbstractIn this paper, we deal with the Nash differential games of stochastic singular systems governed by Itô-type equation in finite-time horizon and infinite-time horizon, respectively. Firstly, the Nash differential game problem of stochastic singular systems in finite time horizon is formulated. By applying the results of stochastic optimal control problem, the existence condition of the Nash strategy is presented by means of a set of cross-coupled Riccati differential equations. Similarly, under the assumption of the admissibility of the stochastic singular systems, the existence condition of the Nash strategy in infinite-time horizon is presented by means of a set of cross-coupled Riccati algebraic equations. The results show that the strategies of each players interact.

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13

Sun, Huiying, Meng Li, Shenglin Ji, and Long Yan. "Stability and Linear Quadratic Differential Games of Discrete-Time Markovian Jump Linear Systems with State-Dependent Noise." Mathematical Problems in Engineering 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/265621.

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We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linear quadratic (LQ) differential games. A necessary and sufficient condition involved with the connection between stochasticTn-stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theory of stochasticTn-stability, we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential games. It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupled generalized algebraic Riccati equations (GAREs). Finally, an iterative algorithm is presented to solve the four coupled GAREs and a simulation example is given to illustrate the effectiveness of it.

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14

Thuan, Do Duc, Nguyen Hong Son, and Cao Thanh Tinh. "Stability radii of differential–algebraic equations with respect to stochastic perturbations." Systems & Control Letters 147 (January 2021): 104834. http://dx.doi.org/10.1016/j.sysconle.2020.104834.

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15

Misawa, Tetsuya. "A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations." SIAM Journal on Scientific Computing 23, no. 3 (January 2001): 866–90. http://dx.doi.org/10.1137/s106482750037024x.

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16

Haider, Michael, and Johannes A. Russer. "Differential form representation of stochastic electromagnetic fields." Advances in Radio Science 15 (September 21, 2017): 21–28. http://dx.doi.org/10.5194/ars-15-21-2017.

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Abstract. In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme based on the Method of Moments (MoM) is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.

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17

KERNER, RICHARD. "STOCHASTIC DESCRIPTION OF AGGLOMERATION AND GROWTH PROCESSES IN GLASSES." International Journal of Modern Physics B 16, no. 14n15 (June 20, 2002): 1987–94. http://dx.doi.org/10.1142/s0217979202011718.

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We show how growth by agglomeration can be described by means of algebraic or differential equations which determine the evolution of probabilities of various local configurations. The minimal fluctuation condition is used to define vitrification. Our methods have been successfully used for the description of glass formation.

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18

Winkler, Renate. "Stochastic differential algebraic equations of index 1 and applications in circuit simulation." Journal of Computational and Applied Mathematics 157, no. 2 (August 2003): 477–505. http://dx.doi.org/10.1016/s0377-0427(03)00436-9.

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19

Milano, Federico, and Rafael Zarate-Minano. "A Systematic Method to Model Power Systems as Stochastic Differential Algebraic Equations." IEEE Transactions on Power Systems 28, no. 4 (November 2013): 4537–44. http://dx.doi.org/10.1109/tpwrs.2013.2266441.

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20

Winkler, Renate. "Stochastic differential algebraic equations of index 1 and applications in circuit simulation." Journal of Computational and Applied Mathematics 163, no. 2 (February 2004): 435–63. http://dx.doi.org/10.1016/j.cam.2003.12.017.

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21

Fahrenwaldt, Matthias A. "Short-time asymptotic expansions of semilinear evolution equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 1 (January 7, 2016): 141–67. http://dx.doi.org/10.1017/s0308210515000372.

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We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

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22

Dragan, Vasile. "On the Linear Quadratic Optimal Control for Systems Described by Singularly Perturbed Itô Differential Equations with Two Fast Time Scales." Axioms 8, no. 1 (March 5, 2019): 30. http://dx.doi.org/10.3390/axioms8010030.

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In this paper a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed Itô differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution (satisfying a prescribed sign condition) to the corresponding stochastic algebraic Riccati equation is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed.

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23

Balaji, S. "Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/304745.

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A Legendre wavelet operational matrix method (LWM) is presented for the solution of nonlinear fractional-order Riccati differential equations, having variety of applications in quantum chemistry and quantum mechanics. The fractional-order Riccati differential equations converted into a system of algebraic equations using Legendre wavelet operational matrix. Solutions given by the proposed scheme are more accurate and reliable and they are compared with recently developed numerical, analytical, and stochastic approaches. Comparison shows that the proposed LWM approach has a greater performance and less computational effort for getting accurate solutions. Further existence and uniqueness of the proposed problem are given and moreover the condition of convergence is verified.

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24

Hirpara, Ravish Himmatlal, and Shambhu Nath Sharma. "An Analysis of a Wind Turbine-Generator System in the Presence of Stochasticity and Fokker-Planck Equations." International Journal of System Dynamics Applications 9, no. 1 (January 2020): 18–43. http://dx.doi.org/10.4018/ijsda.2020010102.

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In power systems dynamics and control literature, theoretical and practical aspects of the wind turbine-generator system have received considerable attentions. The evolution equation of the induction machine encompasses a system of three first-order differential equations coupled with two algebraic equations. After accounting for stochasticity in the wind speed, the wind turbine-generator system becomes a stochastic system. That is described by the standard and formal Itô stochastic differential equation. Note that the Itô process is a strong Markov process. The Itô stochasticity of the wind speed is attributed to the Markov modeling of atmospheric turbulence. The article utilizes the Fokker-Planck method, a mathematical stochastic method, to analyse the noise-influenced wind turbine-generator system by doing the following: (i) the authors develop the Fokker-Planck model for the stochastic power system problem considered here; (ii) the Fokker-Planck operator coupled with the Kolmogorov backward operator are exploited to accomplish the noise analysis from the estimation-theoretic viewpoint.

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25

Rosseel, E., T. Boonen, and S. Vandewalle. "Algebraic multigrid for stationary and time-dependent partial differential equations with stochastic coefficients." Numerical Linear Algebra with Applications 15, no. 2-3 (2008): 141–63. http://dx.doi.org/10.1002/nla.568.

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26

Mohammadi, Fakhrodin. "Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets." Boletim da Sociedade Paranaense de Matemática 35, no. 1 (October 26, 2017): 195. http://dx.doi.org/10.5269/bspm.v35i1.28262.

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‎Stochastic fractional differential equations (SFDEs) have been used for modeling many physical problems in the fields of turbulance‎, ‎heterogeneous‎, ‎flows and matrials‎, ‎viscoelasticity and electromagnetic theory‎. ‎In this paper‎, ‎an‎ efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs‎. ‎In ‎this ‎app‎roach‎‎, ‎o‎perational matrices of the second kind Chebyshev wavelets ‎are used ‎for reducing SFDEs to a linear system of algebraic equations that can be solved easily‎. ‎C‎onvergence and error analysis of the proposed method is ‎considered‎.‎ ‎Some numerical examples are performed to confirm the applicability and efficiency of the proposed method‎.

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27

González-Zumba, Andrés, Pedro Fernández-de-Córdoba, Juan-Carlos Cortés, and Volker Mehrmann. "Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems." Mathematics 8, no. 9 (August 20, 2020): 1393. http://dx.doi.org/10.3390/math8091393.

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In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.

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28

Babaei, Afshin, Hossein Jafari, and S. Banihashemi. "A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise." Symmetry 12, no. 6 (June 1, 2020): 904. http://dx.doi.org/10.3390/sym12060904.

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A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

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29

Zhang, Yi, Na Li, and Jianyu Zhang. "Stochastic stability and Hopf bifurcation analysis of a singular bio-economic model with stochastic fluctuations." International Journal of Biomathematics 12, no. 08 (November 2019): 1950083. http://dx.doi.org/10.1142/s1793524519500839.

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In this paper, we study a class of singular stochastic bio-economic models described by differential-algebraic equations due to the influence of economic factors. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. Stochastic stability and Hopf bifurcations can be analytically determined based on the singular boundary theory of diffusion process, the Maximal Lyapunov exponent and the invariant measure theory. The critical value of the stochastic Hopf bifurcation parameter is obtained and the position of Hopf bifurcation drifting with the parameter increase is presented as a result. Practical example is presented to verify the effectiveness of the results.

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30

Ashyralyev, Allaberen, and Ülker Okur. "Stability of Stochastic Partial Differential Equations." Axioms 12, no. 7 (July 24, 2023): 718. http://dx.doi.org/10.3390/axioms12070718.

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In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications.

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31

Cong, Nguyen Dinh, Stefan Siegmund, and Nguyen Thi The. "Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1." Stochastics 86, no. 5 (March 18, 2014): 776–802. http://dx.doi.org/10.1080/17442508.2013.879141.

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32

Jimenez, J. C. "A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations." Applied Mathematics Letters 15, no. 6 (August 2002): 775–80. http://dx.doi.org/10.1016/s0893-9659(02)00041-1.

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33

Winkler, Renate. "Erratum to: “Stochastic differential algebraic equations of index 1 and applications in circuit simulation”." Journal of Computational and Applied Mathematics 163, no. 2 (February 2004): 433. http://dx.doi.org/10.1016/j.cam.2003.11.001.

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34

Küpper, Dominique, Anne Kværnø, and Andreas Rößler. "A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise." BIT Numerical Mathematics 52, no. 2 (September 15, 2011): 437–55. http://dx.doi.org/10.1007/s10543-011-0354-0.

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35

Levin, Alexander. "Deriving Closed-Form Solutions for Gaussian Pricing Models: A Systematic Time-Domain Approach." International Journal of Theoretical and Applied Finance 01, no. 03 (July 1998): 349–76. http://dx.doi.org/10.1142/s0219024998000205.

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A systematic time-domain approach is presented to the derivation of closed-form solutions for interest-rate contingent assets. A financial system "asset — interest rate market" is assumed to follow an any-factor system of linear stochastic differential equations and some piece-wise defined algebraic equations for the payoffs. Closed-form solutions are expressed through the first two statistical moments of the state variables that are proven to satisfy a deterministic linear system of ordinary differential equations. A number of examples are given to illustrate the method's effectiveness. With no restrictions on the number of factors, solutions are derived for randomly amortizing loans and deposits; any European-style swaptions, caps, and floors; conversion options; Asian-style options, etc. A two-factor arbitrage-free Gaussian term structure is introduced and analyzed.

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36

Zhang, Tingting Qin and Chengjian. "A General Class of One-Step Approximation for Index-1 Stochastic Delay-Differential-Algebraic Equations." Journal of Computational Mathematics 37, no. 2 (June 2019): 151–69. http://dx.doi.org/10.4208/jcm.1711-m2016-0810.

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37

Proß, Sabrina, and Bernhard Bachmann. "An Advanced Environment for Hybrid Modeling of Biological Systems Based on Modelica." Journal of Integrative Bioinformatics 8, no. 1 (March 1, 2011): 1–34. http://dx.doi.org/10.1515/jib-2011-152.

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Summary Biological systems are often very complex so that an appropriate formalism is needed for modeling their behavior. Hybrid Petri Nets, consisting of time-discrete Petri Net elements as well as continuous ones, have proven to be ideal for this task. Therefore, a new Petri Net library was implemented based on the object-oriented modeling language Modelica which allows the modeling of discrete, stochastic and continuous Petri Net elements by differential, algebraic and discrete equations. An appropriate Modelica-tool performs the hybrid simulation with discrete events and the solution of continuous differential equations. A special sub-library contains so-called wrappers for specific reactions to simplify the modeling process.The Modelica-models can be connected to Simulink-models for parameter optimization, sensitivity analysis and stochastic simulation in Matlab.The present paper illustrates the implementation of the Petri Net component models, their usage within the modeling process and the coupling between the Modelica-tool Dymola and Matlab/Simulink. The application is demonstrated by modeling the metabolism of Chinese Hamster Ovary Cells.

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38

Hofmann, Norbert, Thomas Müller-Gronbach, and Klaus Ritter. "The Optimal Discretization of Stochastic Differential Equations." Journal of Complexity 17, no. 1 (March 2001): 117–53. http://dx.doi.org/10.1006/jcom.2000.0570.

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39

Schein, O., and G. Denk. "Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits." Journal of Computational and Applied Mathematics 100, no. 1 (November 1998): 77–92. http://dx.doi.org/10.1016/s0377-0427(98)00138-1.

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40

Luo, Mei, Michal Fečkan, Jin-Rong Wang, and Donal O’Regan. "g-Expectation for Conformable Backward Stochastic Differential Equations." Axioms 11, no. 2 (February 14, 2022): 75. http://dx.doi.org/10.3390/axioms11020075.

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In this paper, we study the applications of conformable backward stochastic differential equations driven by Brownian motion and compensated random measure in nonlinear expectation. From the comparison theorem, we introduce the concept of g-expectation and give related properties of g-expectation. In addition, we find that the properties of conformable backward stochastic differential equations can be deduced from the properties of the generator g. Finally, we extend the nonlinear Doob–Meyer decomposition theorem to more general cases.

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41

Xu, Yan, Fushuan Wen, Hongwei Zhao, Minghui Chen, Zeng Yang, and Huiyu Shang. "Stochastic Small Signal Stability of a Power System with Uncertainties." Energies 11, no. 11 (November 1, 2018): 2980. http://dx.doi.org/10.3390/en11112980.

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The ever-increasing penetration of wind power generation and plug-in electric vehicles introduces stochastic continuous disturbances to the power system. This paper proposes an analytical approach to analyze the influence of stochastic continuous disturbances on power system small signal stability. The noise-to-state stability (NSS) and NSS Lyapunov function (NSS-LF) are adopted for stability analysis with respect to the magnitude of uncertainties in a power system. The power system is modeled as a set of stochastic differential equations (SDEs). The supremum of the norm of the covariance is employed to characterize the influence of magnitudes of uncertainties on the power system. Then the relationship between the magnitudes of stochastic variations and probabilistic stability is explicitly identified by NSS. The proposed method can assess the stochastic stability of the power system by checking some algebraic expressions. Hence, it has high computation efficiency compared with the well-established Monte Carlo based method. Besides, since the magnitudes of the stochastic variations are integrated into the definition of the stochastic stability, the proposed method provides theoretical explanations for the impacts of uncertainties.

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42

Zhang, Yue, and Qingling Zhang. "Stability and Bifurcation Analysis of a Singular Delayed Predator-Prey Bioeconomic Model with Stochastic Fluctuations." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/535634.

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This study investigates a singular delayed predator-prey bioeconomic model with stochastic fluctuations, which is described by differential-algebraic equations because of economic factors. The interior equilibrium of the singular delayed predator-prey bioeconomic model switches from being stable to unstable and then back to being stable, with the increase in time delay. The critical values for stability switches and Hopf bifurcations can be analytically determined. Subsequently, the effect of a fluctuating environment on the singular stochastic delayed predator-prey bioeconomic model obtained by introducing Gaussian white noise terms to the aforementioned deterministic model system is discussed. The fluctuation intensity of the population and harvest effort are calculated by Fourier transform method. Numerical simulation results are presented to verify the effectiveness of the conclusions.

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43

Momeni, Mohammad, Mohsen Riahi Beni, Chiara Bedon, Mohammad Amir Najafgholipour, Seyed Mehdi Dehghan, Behtash JavidSharifi, and Mohammad Ali Hadianfard. "Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method." Applied Sciences 11, no. 19 (October 7, 2021): 9307. http://dx.doi.org/10.3390/app11199307.

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The solution of the motion equation for a structural system under prescribed loading and the prediction of the induced accelerations, velocities, and displacements is of special importance in structural engineering applications. In most cases, however, it is impossible to propose an exact analytical solution, as in the case of systems subjected to stochastic input motions or forces. This is also the case of non-linear systems, where numerical approaches shall be taken into account to handle the governing differential equations. The Legendre–Galerkin matrix (LGM) method, in this regard, is one of the basic approaches to solving systems of differential equations. As a spectral method, it estimates the system response as a set of polynomials. Using Legendre’s orthogonal basis and considering Galerkin’s method, this approach transforms the governing differential equation of a system into algebraic polynomials and then solves the acquired equations which eventually yield the problem solution. In this paper, the LGM method is used to solve the motion equations of single-degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems. The obtained outputs are compared with methods of exact solution (when available), or with the numerical step-by-step linear Newmark-β method. The presented results show that the LGM method offers outstanding accuracy.

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44

Allen, E. J. "Stochastic difference equations and a stochastic partial differential equation for neutron transport." Journal of Difference Equations and Applications 18, no. 8 (August 2012): 1267–85. http://dx.doi.org/10.1080/10236198.2010.488229.

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Giles, Michael B., Mario Hefter, Lukas Mayer, and Klaus Ritter. "Random bit multilevel algorithms for stochastic differential equations." Journal of Complexity 54 (October 2019): 101395. http://dx.doi.org/10.1016/j.jco.2019.01.002.

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46

Batiha, Iqbal M., Ahmad A. Abubaker, Iqbal H. Jebril, Suha B. Al-Shaikh, and Khaled Matarneh. "A Numerical Approach of Handling Fractional Stochastic Differential Equations." Axioms 12, no. 4 (April 17, 2023): 388. http://dx.doi.org/10.3390/axioms12040388.

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This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

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47

Calatayud Gregori, Julia, Benito M. Chen-Charpentier, Juan Carlos Cortés López, and Marc Jornet Sanz. "Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models." Symmetry 11, no. 1 (January 3, 2019): 43. http://dx.doi.org/10.3390/sym11010043.

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In this paper, we deal with computational uncertainty quantification for stochastic models with one random input parameter. The goal of the paper is twofold: First, to approximate the set of probability density functions of the solution stochastic process, and second, to show the capability of our theoretical findings to deal with some important epidemiological models. The approximations are constructed in terms of a polynomial evaluated at the random input parameter, by means of generalized polynomial chaos expansions and the stochastic Galerkin projection technique. The probability density function of the aforementioned univariate polynomial is computed via the random variable transformation method, by taking into account the domains where the polynomial is strictly monotone. The algebraic/exponential convergence of the Galerkin projections gives rapid convergence of these density functions. The examples are based on fundamental epidemiological models formulated via linear and nonlinear differential and difference equations, where one of the input parameters is assumed to be a random variable.

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Kohatsu-Higa, Arturo, Salvador Ortiz-Latorre, and Peter Tankov. "Optimal simulation schemes for Lévy driven stochastic differential equations." Mathematics of Computation 83, no. 289 (December 17, 2013): 2293–324. http://dx.doi.org/10.1090/s0025-5718-2013-02786-x.

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Ali, Ishtiaq, and Sami Ullah Khan. "A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method." Mathematics 11, no. 6 (March 9, 2023): 1328. http://dx.doi.org/10.3390/math11061328.

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This research focuses on the analysis of the competitive model used in the banking sector based on the stochastic fractional differential equation. For the approximate solution, a pseudospectral technique is utilized for the proposed model based on the stochastic Lotka–Volterra equation using a wide range of fractional order parameters in simulations. Conditions for stable and unstable equilibrium points are provided using the Jacobian. The Lotka–Volterra equation is unstable in the long term and can produce highly fluctuating dynamics, which is also one of the reasons that this equation is used to model the problems arising in finance, where fluctuations are important. For this reason, the conventional analytical and numerical methods are not the best choices. To overcome this difficulty, an automatic procedure is used to solve the resultant algebraic equation after the discretization of the operator. In order to fully use the properties of orthogonal polynomials, the proposed scheme is applied to the equivalent integral form of stochastic fractional differential equations under consideration. This also helps in the analysis of fractional differential equations, which mostly fall in the framework of their integrated form. We demonstrate that this fractional approach may be considered as the best tool to model such real-world data situations with very reasonable accuracy. Our numerical simulations further demonstrate that the use of the fractional Atangana–Baleanu operator approach produces results that are more precise and flexible, allowing individuals or companies to use it with confidence to model such real-world situations. It is shown that our numerical simulation results have a very good agreement with the real data, further showing the efficiency and effectiveness of our numerical scheme for the proposed model.

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Hamerle, Alfred, Willi Nagl, and Hermann Singer. "Problems with the estimation of stochastic differential equations using structural equations models." Journal of Mathematical Sociology 16, no. 3 (November 1991): 201–20. http://dx.doi.org/10.1080/0022250x.1991.9990088.

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