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Journal articles on the topic 'Stochastic ordinary differential equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Ünal, Gazanfer. "Stochastic symmetries of Wick type stochastic ordinary differential equations." International Journal of Modern Physics: Conference Series 38 (January 2015): 1560079. http://dx.doi.org/10.1142/s2010194515600794.

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We consider Wick type stochastic ordinary differential equations with Gaussian white noise. We define the stochastic symmetry transformations and Lie equations in Kondratiev space [Formula: see text]. We derive the determining system of Wick type stochastic partial differential equations with Gaussian white noise. Stochastic symmetries for stochastic Bernoulli, Riccati and general stochastic linear equation in [Formula: see text] are obtained. A stochastic version of canonical variables is also introduced.

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2

Deimling, K. "Sample solutions of stochastic ordinary differential equations∗." Stochastic Analysis and Applications 3, no. 1 (January 1985): 15–21. http://dx.doi.org/10.1080/07362998508809051.

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3

Ubøe, Jan. "Measurements of ordinary and stochastic differential equations." Stochastic Processes and their Applications 89, no. 2 (October 2000): 315–31. http://dx.doi.org/10.1016/s0304-4149(00)00026-0.

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4

Just, Wolfram, and Herwig Sauermann. "Ordinary differential equations for nonlinear stochastic oscillators." Physics Letters A 131, no. 4-5 (August 1988): 234–38. http://dx.doi.org/10.1016/0375-9601(88)90018-7.

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5

Luo, Peng, and Falei Wang. "Stochastic differential equations driven by G-Brownian motion and ordinary differential equations." Stochastic Processes and their Applications 124, no. 11 (November 2014): 3869–85. http://dx.doi.org/10.1016/j.spa.2014.07.004.

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6

Hiroshi, Kunita. "Convergence of stochastic flows connected with stochastic ordinary differential equations." Stochastics 17, no. 3 (May 1986): 215–51. http://dx.doi.org/10.1080/17442508608833391.

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7

Papanicolaou, G. C., and W. Kohler. "Asymptotic theory of mixing stochastic ordinary differential equations." Communications on Pure and Applied Mathematics 27, no. 5 (September 13, 2010): 641–68. http://dx.doi.org/10.1002/cpa.3160270503.

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8

Spigler, Renato. "Numerical simulation for certain stochastic ordinary differential equations." Journal of Computational Physics 74, no. 1 (January 1988): 244–62. http://dx.doi.org/10.1016/0021-9991(88)90079-4.

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9

Augustin, F., and P. Rentrop. "Stochastic Galerkin techniques for random ordinary differential equations." Numerische Mathematik 122, no. 3 (April 18, 2012): 399–419. http://dx.doi.org/10.1007/s00211-012-0466-8.

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10

Fredericks, E., and F. M. Mahomed. "Symmetries of th-Order Approximate Stochastic Ordinary Differential Equations." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/263570.

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Symmetries of th-order approximate stochastic ordinary differential equations (SODEs) are studied. The determining equations of these SODEs are derived in an Itô calculus context. These determining equations are not stochastic in nature. SODEs are normally used to model nature (e.g., earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.

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11

Fujiwara, Tsukasa, and Hiroshi Kunita. "Limit theorems for stochastic difference-differential equations." Nagoya Mathematical Journal 127 (September 1992): 83–116. http://dx.doi.org/10.1017/s0027763000004116.

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There are extensive works on the limit theorems for sequences of stochastic ordinary differential equations written in the form:where is a stochastic process and is a deterministic function, both of which take values in the space of vector fields. The case where {ftn} n satisfies certain mixing conditions has been studied by Khas’minskii [7], Kesten-Papanicolaou [6] and others.

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12

Kloeden, Peter E., and Arnulf Jentzen. "Pathwise convergent higher order numerical schemes for random ordinary differential equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2087 (August 21, 2007): 2929–44. http://dx.doi.org/10.1098/rspa.2007.0055.

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Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

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13

Fredericks, E., and F. M. Mahomed. "Symmetries of first-order stochastic ordinary differential equations revisited." Mathematical Methods in the Applied Sciences 30, no. 16 (2007): 2013–25. http://dx.doi.org/10.1002/mma.942.

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14

Fierro, Raul, and Soledad Torres. "A stochastic scheme of approximation for ordinary differential equations." Electronic Communications in Probability 13 (2008): 1–9. http://dx.doi.org/10.1214/ecp.v13-1341.

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15

Higham, D. J. "Stochastic ordinary differential equations in applied and computational mathematics." IMA Journal of Applied Mathematics 76, no. 3 (April 16, 2011): 449–74. http://dx.doi.org/10.1093/imamat/hxr016.

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16

Khan, Junaid Ali, Muhammad Asif Zahoor Raja, and Ijaz Mansoor Qureshi. "Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations." Chinese Physics Letters 28, no. 2 (February 2011): 020206. http://dx.doi.org/10.1088/0256-307x/28/2/020206.

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17

WU, JING. "WIENER–POISSON TYPE MULTIVALUED STOCHASTIC EVOLUTION EQUATIONS IN BANACH SPACES." Stochastics and Dynamics 12, no. 02 (April 8, 2012): 1150015. http://dx.doi.org/10.1142/s0219493712003687.

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We prove the existence and uniqueness of solutions to Wiener–Poisson type multivalued stochastic evolution equations in abstract spaces. We also prove that the solution has the Markov property. Moreover, applications to stochastic ordinary differential equations and stochastic partial differential equations are presented.

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18

Kolarova, Edita, and Lubomir Brancik. "Confidence intervals for RLCG cell influenced by coloured noise." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 36, no. 4 (July 3, 2017): 838–49. http://dx.doi.org/10.1108/compel-07-2016-0321.

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Purpose The purpose of this paper is to determine confidence intervals for the stochastic solutions in RLCG cells with a potential source influenced by coloured noise. Design/methodology/approach The deterministic model of the basic RLCG cell leads to an ordinary differential equation. In this paper, a stochastic model is formulated and the corresponding stochastic differential equation is analysed using the Itô stochastic calculus. Findings Equations for the first and the second moment of the stochastic solution of the coloured noise-affected RLCG cell are obtained, and the corresponding confidence intervals are determined. The moment equations lead to ordinary differential equations, which are solved numerically by an implicit Euler scheme, which turns out to be very effective. For comparison, the confidence intervals are computed statistically by an implementation of the Euler scheme using stochastic differential equations. Practical implications/implications The theoretical results are illustrated by examples. Numerical simulations in the examples are carried out using Matlab. A possible generalization for transmission line models is indicated. Originality/value The Itô-type stochastic differential equation describing the coloured noise RLCG cell is formulated, and equations for the respective moments are derived. Owing to this original approach, the confidence intervals can be found more effectively by solving a system of ordinary differential equations rather than by using statistical methods.

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19

Tornøe, Christoffer W., Rune V. Overgaard, Henrik Agersø, Henrik A. Nielsen, Henrik Madsen, and E. Niclas Jonsson. "Stochastic Differential Equations in NONMEM®: Implementation, Application, and Comparison with Ordinary Differential Equations." Pharmaceutical Research 22, no. 8 (August 2005): 1247–58. http://dx.doi.org/10.1007/s11095-005-5269-5.

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20

Stefanakis, Nikolaos, Markus Abel, and André Bergner. "Sound Synthesis Based on Ordinary Differential Equations." Computer Music Journal 39, no. 3 (September 2015): 46–58. http://dx.doi.org/10.1162/comj_a_00314.

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Ordinary differential equations (ODEs) have been studied for centuries as a means to model complex dynamical processes from the real world. Nevertheless, their application to sound synthesis has not yet been fully exploited. In this article we present a systematic approach to sound synthesis based on first-order complex and real ODEs. Using simple time-dependent and nonlinear terms, we illustrate the mapping between ODE coefficients and physically meaningful control parameters such as pitch, pitch bend, decay rate, and attack time. We reveal the connection between nonlinear coupling terms and frequency modulation, and we discuss the implications of this scheme in connection with nonlinear synthesis. The ability to excite a first-order complex ODE with an external input signal is also examined; stochastic or impulsive signals that are physically or synthetically produced can be presented as input to the system, offering additional synthesis possibilities, such as those found in excitation/filter synthesis and filter-based modal synthesis.

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21

Heredia, Jorge Pérez. "Modelling Evolutionary Algorithms with Stochastic Differential Equations." Evolutionary Computation 26, no. 4 (December 2018): 657–86. http://dx.doi.org/10.1162/evco_a_00216.

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There has been renewed interest in modelling the behaviour of evolutionary algorithms (EAs) by more traditional mathematical objects, such as ordinary differential equations or Markov chains. The advantage is that the analysis becomes greatly facilitated due to the existence of well established methods. However, this typically comes at the cost of disregarding information about the process. Here, we introduce the use of stochastic differential equations (SDEs) for the study of EAs. SDEs can produce simple analytical results for the dynamics of stochastic processes, unlike Markov chains which can produce rigorous but unwieldy expressions about the dynamics. On the other hand, unlike ordinary differential equations (ODEs), they do not discard information about the stochasticity of the process. We show that these are especially suitable for the analysis of fixed budget scenarios and present analogues of the additive and multiplicative drift theorems from runtime analysis. In addition, we derive a new more general multiplicative drift theorem that also covers non-elitist EAs. This theorem simultaneously allows for positive and negative results, providing information on the algorithm's progress even when the problem cannot be optimised efficiently. Finally, we provide results for some well-known heuristics namely Random Walk (RW), Random Local Search (RLS), the (1+1) EA, the Metropolis Algorithm (MA), and the Strong Selection Weak Mutation (SSWM) algorithm.

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22

Carletti, Margherita, and Malay Banerjee. "A Backward Technique for Demographic Noise in Biological Ordinary Differential Equation Models." Mathematics 7, no. 12 (December 9, 2019): 1204. http://dx.doi.org/10.3390/math7121204.

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Physical systems described by deterministic differential equations represent idealized situations since they ignore stochastic effects. In the context of biomathematical modeling, we distinguish between environmental or extrinsic noise and demographic or intrinsic noise, for which it is assumed that the variation over time is due to demographic variation of two or more interacting populations (birth, death, immigration, and emigration). The modeling and simulation of demographic noise as a stochastic process affecting units of populations involved in the model is well known in the literature, resulting in discrete stochastic systems or, when the population sizes are large, in continuous stochastic ordinary differential equations and, if noise is ignored, in continuous ordinary differential equation models. The inverse process, i.e., inferring the effects of demographic noise on a natural system described by a set of ordinary differential equations, is still an issue to be addressed. With this paper, we provide a technique to model and simulate demographic noise going backward from a deterministic continuous differential system to its underlying discrete stochastic process, based on the framework of chemical kinetics, since demographic noise is nothing but the biological or ecological counterpart of intrinsic noise in genetic regulation. Our method can, thus, be applied to ordinary differential systems describing any kind of phenomena when intrinsic noise is of interest.

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23

Carletti, M., K. Burrage, and P. M. Burrage. "Numerical simulation of stochastic ordinary differential equations in biomathematical modelling." Mathematics and Computers in Simulation 64, no. 2 (January 2004): 271–77. http://dx.doi.org/10.1016/j.matcom.2003.09.022.

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24

Soh, C. Wafo, and F. M. Mahomed. "Integration of stochastic ordinary differential equations from a symmetry standpoint." Journal of Physics A: Mathematical and General 34, no. 1 (December 19, 2000): 177–92. http://dx.doi.org/10.1088/0305-4470/34/1/314.

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25

Radu, Alin. "Stochastic reduced-order models for stable nonlinear ordinary differential equations." Nonlinear Dynamics 97, no. 1 (May 7, 2019): 225–45. http://dx.doi.org/10.1007/s11071-019-04967-x.

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26

Spigler, Renato. "Monte Carlo-type simulation for solving stochastic ordinary differential equations." Mathematics and Computers in Simulation 29, no. 3-4 (July 1987): 243–51. http://dx.doi.org/10.1016/0378-4754(87)90134-0.

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27

Buckwar, Evelyn, and Renate Winkler. "Improved linear multi-step methods for stochastic ordinary differential equations." Journal of Computational and Applied Mathematics 205, no. 2 (August 2007): 912–22. http://dx.doi.org/10.1016/j.cam.2006.03.038.

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28

Cheban, David, and Zhenxin Liu. "Averaging principle on infinite intervals for stochastic ordinary differential equations." Electronic Research Archive 29, no. 4 (2021): 2791. http://dx.doi.org/10.3934/era.2021014.

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29

Li, Ning, Bo Meng, Xinlong Feng, and Dongwei Gui. "A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos." East Asian Journal on Applied Mathematics 5, no. 2 (May 2015): 192–208. http://dx.doi.org/10.4208/eajam.250714.020515a.

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AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

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30

BUCKWAR, E., M. G. RIEDLER, and P. E. KLOEDEN. "THE NUMERICAL STABILITY OF STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE." Stochastics and Dynamics 11, no. 02n03 (September 2011): 265–81. http://dx.doi.org/10.1142/s0219493711003279.

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An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

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31

Gliklikh, Yuri E., and Lora A. Morozova. "Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations." International Journal of Mathematics and Mathematical Sciences 2004, no. 17 (2004): 901–12. http://dx.doi.org/10.1155/s016117120430503x.

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First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one-sided estimate type that is formulated in terms of so-called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts:L1-complete solutions of SDE (this means that they belong to a certain functional space ofL1type) and the so-called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

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32

RUPŠYS, PETRAS, and EDMUNDAS PETRAUSKAS. "ANALYSIS OF HEIGHT CURVES BY STOCHASTIC DIFFERENTIAL EQUATIONS." International Journal of Biomathematics 05, no. 05 (June 17, 2012): 1250045. http://dx.doi.org/10.1142/s1793524511001878.

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Height–diameter models are classically analyzed by fixed or mixed linear and non-linear regression models. In order to possess the among-plot variability, we propose the methodology of stochastic differential equations that is derived from the standard deterministic ordinary differential equation by adding the process variability to the growth dynamic. Age–diameter varying height model was deduced using a two-dimensional stochastic Gompertz shape process. Another focus of the article is the investigation of normal copula procedure, when the tree diameter and height are governed by univariate stochastic Gompertz shape processes. The advantage of the stochastic differential equation methodology is that it analyzes a residual variability, corresponding to measurements error, and an individual variability to represent heterogeneity between subjects more complex than commonly used fixed effect models. An analysis of 900 Scots pine (Pinus sylvestris) trees provided the data for this study.

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33

Li, Changzhao, and Hui Fang. "Stochastic Bifurcations of Group-Invariant Solutions for a Generalized Stochastic Zakharov–Kuznetsov Equation." International Journal of Bifurcation and Chaos 31, no. 03 (March 15, 2021): 2150040. http://dx.doi.org/10.1142/s0218127421500401.

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In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic bifurcations of group-invariant solutions for a generalized stochastic Zakharov–Kuznetsov equation.

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34

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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35

Platen, Eckhard. "An introduction to numerical methods for stochastic differential equations." Acta Numerica 8 (January 1999): 197–246. http://dx.doi.org/10.1017/s0962492900002920.

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This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

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36

Bell, Denis R., and Salah E. A. Mohammed. "On the solution of stochastic ordinary differential equations via small delays." Stochastics and Stochastic Reports 28, no. 4 (December 1989): 293–99. http://dx.doi.org/10.1080/17442508908833598.

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37

Alcock, Jamie, and Kevin Burrage. "Stable strong order 1.0 schemes for solving stochastic ordinary differential equations." BIT Numerical Mathematics 52, no. 3 (February 7, 2012): 539–57. http://dx.doi.org/10.1007/s10543-012-0372-6.

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38

Guangping, Luo, Liang Juan, and Zhu Changrong. "The transversal homoclinic solutions and chaos for stochastic ordinary differential equations." Journal of Mathematical Analysis and Applications 412, no. 1 (April 2014): 301–25. http://dx.doi.org/10.1016/j.jmaa.2013.10.055.

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39

Milošević, Marija. "Divergence of the backward Euler method for ordinary stochastic differential equations." Numerical Algorithms 82, no. 4 (January 11, 2019): 1395–407. http://dx.doi.org/10.1007/s11075-019-00661-6.

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40

Alcock, Jamie, and Kevin Burrage. "A genetic estimation algorithm for parameters of stochastic ordinary differential equations." Computational Statistics & Data Analysis 47, no. 2 (September 2004): 255–75. http://dx.doi.org/10.1016/j.csda.2003.11.025.

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41

Buckwar, Evelyn, Rachel Kuske, Salah-Eldin Mohammed, and Tony Shardlow. "Weak Convergence of the Euler Scheme for Stochastic Differential Delay Equations." LMS Journal of Computation and Mathematics 11 (2008): 60–99. http://dx.doi.org/10.1112/s146115700000053x.

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AbstractWe study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

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42

Uslu, Hande, Murat Sari, and Tahir Cosgun. "Qualitative behavior of stiff ODEs through a stochastic approach." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 10, no. 2 (June 4, 2020): 181–87. http://dx.doi.org/10.11121/ijocta.01.2020.00829.

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In the last few decades, stiff differential equations have attracted a great deal of interest from academic society, because much of the real life is covered by stiff behavior. In addition to importance of producing model equations, capturing an exact behavior of the problem by dealing with a solution method is also handling issue. Although there are many explicit and implicit numerical methods for solving them, those methods cannot be properly applied due to their computational time, computational error or effort spent for construction of a structure. Therefore, simulation techniques can be taken into account in capturing the stiff behavior. In this respect, this study aims at analyzing stiff processes through stochastic approaches. Thus, a Monte Carlo based algorithm has been presented for solving some stiff ordinary differential equations and system of stiff linear ordinary differential equations. The produced results have been qualitatively and quantitatively discussed.

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43

Odekunle, M., M. Egwurube, K. Joshua, and A. Adesanya. "Two-Stage Explicit Stochastic Rational Runge-Kutta Method for Solving Stochastic Ordinary Differential Equations." British Journal of Mathematics & Computer Science 12, no. 3 (January 10, 2016): 1–11. http://dx.doi.org/10.9734/bjmcs/2016/18893.

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44

El Fatini, Mohamed, Mohammed Louriki, Roger Pettersson, and Zarife Zararsiz. "Epidemic modeling: Diffusion approximation vs. stochastic differential equations allowing reflection." International Journal of Biomathematics 14, no. 05 (March 5, 2021): 2150036. http://dx.doi.org/10.1142/s1793524521500364.

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A birth–death process is considered as an epidemic model with recovery and transmittance from outside. The fraction of infected individuals is for huge population sizes approximated by a solution of an ordinary differential equation taking values in [Formula: see text]. For intermediate size or semilarge populations, the fraction of infected individuals is approximated by a diffusion formulated as a stochastic differential equation. That diffusion approximation however needs to be killed at the boundary [Formula: see text]. An alternative stochastic differential equation model is investigated which instead allows a more natural reflection at the boundary.

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45

Hasal, Pavel, and Vladimír Kudrna. "Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Numerical Simulation of One-Dimensional Diffusion Process." Collection of Czechoslovak Chemical Communications 61, no. 4 (1996): 512–35. http://dx.doi.org/10.1135/cccc19960512.

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Some problems are analyzed arising when a numerical simulation of a random motion of a large ensemble of diffusing particles is used to approximate the solution of a one-dimensional diffusion equation. The particle motion is described by means of a stochastic differential equation. The problems emerging especially when the diffusion coefficient is a function of spatial coordinate are discussed. The possibility of simulation of various kinds of stochastic integral is demonstrated. It is shown that the application of standard numerical procedures commonly adopted for ordinary differential equations may lead to erroneous results when used for solution of stochastic differential equations. General conclusions are verified by numerical solution of three stochastic differential equations with different forms of the diffusion coefficient.

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46

Delgado-Vences, Francisco, and Franco Flandoli. "A spectral-based numerical method for Kolmogorov equations in Hilbert spaces." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 03 (August 31, 2016): 1650020. http://dx.doi.org/10.1142/s021902571650020x.

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We propose a numerical solution for the solution of the Fokker–Planck–Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein–Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener–Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher–KPP stochastic equation and a stochastic Burgers equation in dimension 1.

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47

IMKELLER, PETER, and CHRISTIAN LEDERER. "THE COHOMOLOGY OF STOCHASTIC AND RANDOM DIFFERENTIAL EQUATIONS, AND LOCAL LINEARIZATION OF STOCHASTIC FLOWS." Stochastics and Dynamics 02, no. 02 (June 2002): 131–59. http://dx.doi.org/10.1142/s021949370200039x.

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Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

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48

Barrera, Antonio, Patricia Román-Román, and Francisco Torres-Ruiz. "Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior." Mathematics 8, no. 2 (January 22, 2020): 155. http://dx.doi.org/10.3390/math8020155.

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Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is addressed and a strategy is established for this purpose. Finally, a simulation study is carried out.

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49

Iizuka, Masaru. "Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation." Journal of Mathematical Biology 25, no. 6 (December 1987): 643–52. http://dx.doi.org/10.1007/bf00275500.

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50

Krebs, Johannes T. N. "Consistency and asymptotic normality of stochastic Euler schemes for ordinary differential equations." Statistics & Probability Letters 125 (June 2017): 1–8. http://dx.doi.org/10.1016/j.spl.2017.01.016.

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