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Journal articles on the topic 'Numerical scheme for SDEs'

Author: Grafiati

Published: 9 September 2022

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1

C. De Vecchi, Francesco, Andrea Romano, and Stefania Ugolini. "A symmetry-adapted numerical scheme for SDEs." Journal of Geometric Mechanics 11, no. 3 (2019): 325–59. http://dx.doi.org/10.3934/jgm.2019018.

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2

Yamada, Toshihiro. "High order weak approximation for irregular functionals of time-inhomogeneous SDEs." Monte Carlo Methods and Applications 27, no. 2 (February 20, 2021): 117–36. http://dx.doi.org/10.1515/mcma-2021-2085.

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Abstract This paper shows a general weak approximation method for time-inhomogeneous stochastic differential equations (SDEs) using Malliavin weights. A unified approach is introduced to construct a higher order discretization scheme for expectations of non-smooth functionals of solutions of time-inhomogeneous SDEs. Numerical experiments show the validity of the method.

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3

Ewald, Brian. "Weak Versions of Stochastic Adams-Bashforth and Semi-implicit Leapfrog Schemes for SDEs." Computational Methods in Applied Mathematics 12, no. 1 (2012): 23–31. http://dx.doi.org/10.2478/cmam-2012-0002.

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AbstractWe consider the weak analogues of certain strong stochastic numerical schemes, namely an Adams-Bashforth scheme and a semi-implicit leapfrog scheme. We show that the weak version of the Adams-Bashforth scheme converges weakly with order 2, and the weak version of the semi-implicit leapfrog scheme converges weakly with order 1. We also note that the weak schemes are computationally simpler and easier to implement than the corresponding strong schemes, resulting in savings in both programming and computational effort.

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4

Li, Xingjie Helen, Fei Lu, and Felix X. F. Ye. "ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems." Discrete & Continuous Dynamical Systems - S 15, no. 4 (2022): 747. http://dx.doi.org/10.3934/dcdss.2021103.

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<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>

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5

Armstrong, J., and D. Brigo. "Intrinsic stochastic differential equations as jets." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2210 (February 2018): 20170559. http://dx.doi.org/10.1098/rspa.2017.0559.

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We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.

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6

Mao, Xuerong, Aubrey Truman, and Chenggui Yuan. "Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching." Journal of Applied Mathematics and Stochastic Analysis 2006 (July 13, 2006): 1–20. http://dx.doi.org/10.1155/jamsa/2006/80967.

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Stochastic differential equations (SDEs) under regime-switching have recently been developed to model various financial quantities. In general, SDEs under regime-switching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation of financial quantities. In this paper, we will concentrate on the Euler-Maruyama (EM) scheme for the typical hybrid mean-reverting θ-process. To overcome the mathematical difficulties arising from the regime-switching as well as the non-Lipschitz coefficients, several new techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.

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7

Zhang, Wei. "Ergodic SDEs on submanifolds and related numerical sampling schemes." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (February 12, 2020): 391–430. http://dx.doi.org/10.1051/m2an/2019071.

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In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure μ on the level set of a smooth function ξ : ℝd → ℝk, 1 ≤ k < d. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff’s ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is μ. Motivated by the previous work of Ciccotti et al. (Commun. Pur. Appl. Math. 61 (2008) 371–408), as well as the work of Leliévre et al. (Math. Comput. 81 (2012) 2071–2125), in this paper we construct a family of ergodic diffusion processes on the level set of ξ whose invariant measures coincide with the given one. For the conditional measure, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn’t require computing the second derivatives of ξ and the error estimates of its long time sampling efficiency are obtained.

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8

Buckwar, Evelyn, Massimiliano Tamborrino, and Irene Tubikanec. "Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs." Statistics and Computing 30, no. 3 (November 5, 2019): 627–48. http://dx.doi.org/10.1007/s11222-019-09909-6.

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Abstract Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g. the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography data. The derived summaries are particularly robust to the model simulation, and this fact, combined with the proposed reliable numerical scheme, yields accurate ABC inference. In contrast, the inference returned using standard numerical methods (Euler–Maruyama discretisation) fails. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.

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9

BRUTI-LIBERATI, NICOLA, and ECKHARD PLATEN. "STRONG PREDICTOR–CORRECTOR EULER METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS." Stochastics and Dynamics 08, no. 03 (September 2008): 561–81. http://dx.doi.org/10.1142/s0219493708002457.

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This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor–corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor–corrector Euler methods.

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10

Kloeden, P. E., and S. Shott. "Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs." Journal of Applied Mathematics and Stochastic Analysis 14, no. 1 (January 1, 2001): 47–53. http://dx.doi.org/10.1155/s1048953301000053.

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Linear-implicit versions of strong Taylor numerical schemes for finite dimensional Itô stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an γ strong linear-implicit Taylor scheme with time-step Δ applied to the N dimensional Itô-Galerkin SDE for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1≤λ2≤… in its drift term is then estimated by K(λN+1−½+Δγ) where the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.

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11

Hurn, Stan, Kenneth A. Lindsay, and Lina Xu. "Revisiting the numerical solution of stochastic differential equations." China Finance Review International 9, no. 3 (August 19, 2019): 312–23. http://dx.doi.org/10.1108/cfri-12-2018-0155.

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Purpose The purpose of this paper is to revisit the numerical solutions of stochastic differential equations (SDEs). An important drawback when integrating SDEs numerically is the number of steps required to attain acceptable accuracy of convergence to the true solution. Design/methodology/approach This paper develops a bias reducing method based loosely on extrapolation. Findings The method is seen to perform acceptably well and for realistic steps sizes provides improved accuracy at no significant additional computational cost. In addition, the optimal step size of the bias reduction methods is shown to be consistent with theoretical analysis. Originality/value Overall, there is evidence to suggest that the proposed method is a viable, easy to implement competitor for other commonly used numerical schemes.

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12

Grigo, Alexander. "High-order numerical schemes for jump-SDEs." Journal of Computational and Applied Mathematics 354 (July 2019): 31–38. http://dx.doi.org/10.1016/j.cam.2018.12.018.

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13

Hodyss, Daniel, Kevin C. Viner, Alex Reinecke, and James A. Hansen. "The Impact of Noisy Physics on the Stability and Accuracy of Physics–Dynamics Coupling." Monthly Weather Review 141, no. 12 (November 25, 2013): 4470–86. http://dx.doi.org/10.1175/mwr-d-13-00035.1.

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Abstract The coupling of the dynamical core of a numerical weather prediction model to the physical parameterizations is an important component of model design. This coupling between the physics and the dynamics is explored here from the perspective of stochastic differential equations (SDEs). It will be shown that the basic properties of the impact of noisy physics on the stability and accuracy of common numerical methods may be obtained through the application of the basic principles of SDEs. A conceptual model setting is used that allows the study of the impact of noise whose character may be tuned to be either very red (smooth) or white (noisy). The change in the stability and accuracy of common numerical methods as the character of the noise changes is then studied. Distinct differences are found between the ability of multistage (Runge–Kutta) schemes as compared with multistep (Adams–Bashforth/leapfrog) schemes to handle noise of various characters. These differences will be shown to be attributable to the basic philosophy used to design the scheme. Additional experiments using the decentering of the noisy physics will also be shown to lead to strong sensitivity to the quality of the noise. As an example, the authors find the novel result that noise of a diffusive character may lead to instability when the scheme is decentered toward greater implicitness. These results are confirmed in a nonlinear shear layer simulation using a subgrid-scale mixing parameterization. This subgrid-scale mixing parameterization is modified stochastically and shown to reproduce the basic principles found here, including the notion that decentering toward implicitness may lead to instability.

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14

Bognash, Mohamed, and Samuel Asokanthan. "Stochastic Stability of a Class of MEMS-Based Vibratory Gyroscopes under Input Rate Fluctuations." Vibration 1, no. 1 (June 19, 2018): 69–80. http://dx.doi.org/10.3390/vibration1010006.

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The influence of stochastic fluctuations in the input angular rate of a class of single axis mass-spring microelectromechanical (MEM) gyroscopes on the system stability is investigated. A white noise fluctuation is introduced in the coupled 2-DOF model that represents the system dynamics for the purposes of stability prediction. Numerical simulations are performed employing the resulting set of stochastic differential equations (SDEs) that govern the system dynamics. The SDEs are discretized using the higher-order Milstein scheme for numerical computations. Simulations via the Euler scheme, as well as the measure of the largest Lyapunov exponent are employed for validation purposes due to a lack of similar analytical solutions or experimental data. Responses have been predicted under different noise fluctuation magnitudes and different input angular rates for stability investigations. A parametric study is performed to estimate the noise intensity stability threshold for a range of quality factor values at different input angular rates. The predicted results show a nonlinear dependence of the threshold on the quality factors for different input rates. Under typical gyroscope operating conditions, a realistic frequency mismatch appears to have insignificant influence on system stability. It is envisaged that the present quantitative predictions will aid improvements in performance, reliability, and the design process for this class of devices.

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15

Ruijter, M. J., and C. W. Oosterlee. "Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance." Applied Numerical Mathematics 103 (May 2016): 1–26. http://dx.doi.org/10.1016/j.apnum.2015.12.003.

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16

Liu, Shuaiqiang, Lech A. Grzelak, and Cornelis W. Oosterlee. "The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations." Risks 10, no. 3 (February 23, 2022): 47. http://dx.doi.org/10.3390/risks10030047.

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We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

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17

Chen, Lin, and Fu Ke Wu. "Almost Sure Decay Stability of the Backward Euler-Maruyama Scheme for Stochastic Differential Equations with Unbounded Delay." Applied Mechanics and Materials 235 (November 2012): 39–44. http://dx.doi.org/10.4028/www.scientific.net/amm.235.39.

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This paper deals with analytical and numerical stability properties of highly nonlinear stochastic differential equations (SDEs) with unbounded delay. Sufficient conditions for almost sure decay stability of previous system, almost sure decay stability of the backward Euler-Maruyama (BEM) methods are investigated. In \cite{Wu2010} and \cite{Mao2011}, the authors consider one-side linear growth condition and sufficient small step size. In this paper, we consider the monotone condition, which is weaker than one-side linear growth condition. And we only need a very weak restriction of the step size. Different from \cite{Szpruch2010}, Szpruch and Mao consider the asymptotic stability of the numerical approximate. In this paper we consider the almost sure decay stability of the numerical solution. This improves the existing results greatly.

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18

Alnafisah, Yousef. "The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method." Abstract and Applied Analysis 2018 (2018): 1–7. http://dx.doi.org/10.1155/2018/3805042.

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Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.

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19

Briand, Phillippe, Abir Ghannoum, and Céline Labart. "Mean reflected stochastic differential equations with jumps." Advances in Applied Probability 52, no. 2 (June 2020): 523–62. http://dx.doi.org/10.1017/apr.2020.11.

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AbstractIn this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

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20

Hutzenthaler, Martin, Arnulf Jentzen, and Peter E. Kloeden. "Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2130 (December 15, 2010): 1563–76. http://dx.doi.org/10.1098/rspa.2010.0348.

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The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.

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21

Okano, Yusuke, and Toshihiro Yamada. "A control variate method for weak approximation of SDEs via discretization of numerical error of asymptotic expansion." Monte Carlo Methods and Applications 25, no. 3 (September 1, 2019): 239–52. http://dx.doi.org/10.1515/mcma-2019-2044.

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Abstract The paper shows a new weak approximation method for stochastic differential equations as a generalization and an extension of Heath–Platen’s scheme for multidimensional diffusion processes. We reformulate the Heath–Platen estimator from the viewpoint of asymptotic expansion. The proposed scheme is implemented by a Monte Carlo method and its variance is much reduced by the asymptotic expansion which works as a kind of control variate. Numerical examples for the local stochastic volatility model are shown to confirm the efficiency of the method.

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22

Naito, Riu, and Toshihiro Yamada. "A second-order discretization for forward-backward SDEs using local approximations with Malliavin calculus." Monte Carlo Methods and Applications 25, no. 4 (December 1, 2019): 341–61. http://dx.doi.org/10.1515/mcma-2019-2053.

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Abstract The paper proposes a new second-order discretization method for forward-backward stochastic differential equations. The method is given by an algorithm with polynomials of Brownian motions where the local approximations using Malliavin calculus play a role. For the implementation, we introduce a new least squares Monte Carlo method for the scheme. A numerical example is illustrated to check the effectiveness.

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23

Lehn, J., A. Rößler, and O. Schein. "Adaptive schemes for the numerical solution of SDEs—a comparison." Journal of Computational and Applied Mathematics 138, no. 2 (January 2002): 297–308. http://dx.doi.org/10.1016/s0377-0427(01)00375-2.

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24

Alhojilan, Yazid. "Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion." Open Mathematics 17, no. 1 (December 31, 2019): 1515–25. http://dx.doi.org/10.1515/math-2019-0124.

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Abstract This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

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25

Ferreiro-Castilla, A., A. E. Kyprianou, and R. Scheichl. "An Euler–Poisson scheme for Lévy driven stochastic differential equations." Journal of Applied Probability 53, no. 1 (March 2016): 262–78. http://dx.doi.org/10.1017/jpr.2015.23.

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Abstract We describe an Euler scheme to approximate solutions of Lévy driven stochastic differential equations (SDEs) where the grid points are given by the arrival times of a Poisson process and thus are random. This result extends the previous work of Ferreiro-Castilla et al. (2014). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and prove that the mean-square error converges with rate O(n-1/2). The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the SDE. Classical examples, such as stable processes, subclasses of spectrally one-sided Lévy processes, and new families, such as meromorphic Lévy processes (Kuznetsov et al. (2012), are examples for which our algorithm provides an interesting alternative to existing methods, due to its straightforward implementation and its robustness with respect to the jump structure of the driving Lévy process.

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26

Ngo, Hoang-Long, and Dai Taguchi. "On the Euler–Maruyama scheme for SDEs with bounded variation and Hölder continuous coefficients." Mathematics and Computers in Simulation 161 (July 2019): 102–12. http://dx.doi.org/10.1016/j.matcom.2019.01.012.

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BENDER, CHRISTIAN, and MICHAEL KOHLMANN. "OPTIMAL SUPERHEDGING UNDER NON-CONVEX CONSTRAINTS — A BSDE APPROACH." International Journal of Theoretical and Applied Finance 11, no. 04 (June 2008): 363–80. http://dx.doi.org/10.1142/s0219024908004841.

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We apply theoretical results by Peng on supersolutions for Backward SDEs (BSDEs) to the problem of finding optimal superhedging strategies in a generalized Black–Scholes market under constraints. Constraints may be imposed simultaneously on wealth process and portfolio. They may be non-convex, time-dependent, and random. The BSDE method turns out to be an extremely useful tool for modeling realistic markets: in this paper, it is shown how more realistic constraints on the portfolio may be formulated via BSDE theory in terms of the amount of money invested, the portfolio proportion, or the number of shares held. Based on recent advances on numerical methods for BSDEs (in particular, the forward scheme by Bender and Denk [1]), a Monte Carlo method for approximating the superhedging price is given, which demonstrates the practical applicability of the BSDE method. Some numerical examples concerning European and American options under non-convex borrowing constraints are presented.

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28

Zhang, Zhongqiang, and Heping Ma. "Order-preserving strong schemes for SDEs with locally Lipschitz coefficients." Applied Numerical Mathematics 112 (February 2017): 1–16. http://dx.doi.org/10.1016/j.apnum.2016.09.013.

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29

JABERI, F. A., P. J. COLUCCI, S. JAMES, P. GIVI, and S. B. POPE. "Filtered mass density function for large-eddy simulation of turbulent reacting flows." Journal of Fluid Mechanics 401 (December 25, 1999): 85–121. http://dx.doi.org/10.1017/s0022112099006643.

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A methodology termed the ‘filtered mass density function’ (FMDF) is developed and implemented for large-eddy simulation (LES) of variable-density chemically reacting turbulent flows at low Mach numbers. This methodology is based on the extension of the ‘filtered density function’ (FDF) scheme recently proposed by Colucci et al. (1998) for LES of constant-density reacting flows. The FMDF represents the joint probability density function of the subgrid-scale (SGS) scalar quantities and is obtained by solution of its modelled transport equation. In this equation, the effect of chemical reactions appears in a closed form and the influences of SGS mixing and convection are modelled. The stochastic differential equations (SDEs) which yield statistically equivalent results to those of the FMDF transport equation are derived and are solved via a Lagrangian Monte Carlo scheme. The consistency, convergence, and accuracy of the FMDF and the Monte Carlo solution of its equivalent SDEs are assessed. In non-reacting flows, it is shown that the filtered results via the FMDF agree well with those obtained by the ‘conventional’ LES in which the finite difference solution of the transport equations of these filtered quantities is obtained. The advantage of the FMDF is demonstrated in LES of reacting shear flows with non-premixed reactants. The FMDF results are appraised by comparisons with data generated by direct numerical simulation (DNS) and with experimental measurements. In the absence of a closure for the SGS scalar correlations, the results based on the conventional LES are significantly different from those obtained by DNS. The FMDF results show a closer agreement with DNS. These results also agree favourably with laboratory data of exothermic reacting turbulent shear flows, and portray several of the features observed experimentally.

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Akahori, Jirô, Masahiro Kinuya, Takashi Sawai, and Tomooki Yuasa. "An efficient weak Euler–Maruyama type approximation scheme of very high dimensional SDEs by orthogonal random variables." Mathematics and Computers in Simulation 187 (September 2021): 540–65. http://dx.doi.org/10.1016/j.matcom.2021.03.010.

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31

Szpruch, Łukasz, and Xīlíng Zhāng. "$V$-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs." Mathematics of Computation 87, no. 310 (August 3, 2017): 755–83. http://dx.doi.org/10.1090/mcom/3219.

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32

Ngo, Ichhuy, Kyuro Sasaki, Liqiang Ma, Ronald Nguele, and Yuichi Sugai. "Enhancing surfactant desorption through low salinity water post-flush during Enhanced Oil Recovery." Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles 76 (2021): 68. http://dx.doi.org/10.2516/ogst/2021050.

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Low Salinity Water (LSW) incorporates in surfactant Enhanced Oil Recovery (EOR) as a pre-flush is a common practice aiming to reduce the formation salinity, which affects surfactant adsorption. However, in a field implementation, the adsorption of surfactant is unavoidable, so creating a scheme that detaches the trapped surfactant is equally essential. In this study, LSW was a candidate to enhance the desorption of surfactant. LSW solely formulated from NaCl (1 wt.%), Sodium Dodecylbenzene Sulfonate (SDBS) was chosen as the primary surfactant at its critical micelle concentration (CMC, 0.1 wt.%). It found that injecting LSW as post-flush achieved up to 71.7% of SDBS desorption that lower interfacial tension against oil (31.06° API) to 1.3 mN/m hence bring the total Recovery Factor (RF) to 56.1%. It was 4.9% higher than when LSW injecting as pre-flush and 5.2% greater than conventional surfactant flooding (without LSW). Chemical analysis unveiled salinity reduction induces Na+ ion adsorption substitution onto pore surface resulting in an increment in surfactant desorption. The study was further conducted in a numerical simulation upon history matched with core-flood data reported previously. By introducing LSW in post-flush after SDBS injection, up to 5.6% RF increased in comparison to other schemes. The proposed scheme resolved the problems of adsorbed surfactant after EOR, and further improve the economic viability of surfactant EOR.

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Bender, Christian, and Robert Denk. "A forward scheme for backward SDEs." Stochastic Processes and their Applications 117, no. 12 (December 2007): 1793–812. http://dx.doi.org/10.1016/j.spa.2007.03.005.

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34

Faizullah, Faiz. "A Note on the Carathéodory Approximation Scheme for Stochastic Differential Equations under G-Brownian Motion." Zeitschrift für Naturforschung A 67, no. 12 (December 1, 2012): 699–704. http://dx.doi.org/10.5560/zna.2012-0079.

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In this note, the Carathéodory approximation scheme for vector valued stochastic differential equations under G-Brownian motion (G-SDEs) is introduced. It is shown that the Carathéodory approximate solutions converge to the unique solution of the G-SDEs. The existence and uniqueness theorem for G-SDEs is established by using the stated method.

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35

Eissa, Mahmoud A., Haiying Zhang, and Yu Xiao. "Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations." Mathematical Problems in Engineering 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/1682513.

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The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

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Higham, Desmond J., Xuerong Mao, and Andrew M. Stuart. "Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations." LMS Journal of Computation and Mathematics 6 (2003): 297–313. http://dx.doi.org/10.1112/s1461157000000462.

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AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

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Kubilius, Kęstutis, and Aidas Medžiūnas. "Pathwise Convergent Approximation for the Fractional SDEs." Mathematics 10, no. 4 (February 21, 2022): 669. http://dx.doi.org/10.3390/math10040669.

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Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

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Li, Nanxi, Hongbo Shi, Bing Song, and Yang Tao. "Temporal-Spatial Neighborhood Enhanced Sparse Autoencoder for Nonlinear Dynamic Process Monitoring." Processes 8, no. 9 (September 1, 2020): 1079. http://dx.doi.org/10.3390/pr8091079.

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Data-based process monitoring methods have received tremendous attention in recent years, and modern industrial process data often exhibit dynamic and nonlinear characteristics. Traditional autoencoders, such as stacked denoising autoencoders (SDAEs), have excellent nonlinear feature extraction capabilities, but they ignore the dynamic correlation between sample data. Feature extraction based on manifold learning using spatial or temporal neighbors has been widely used in dynamic process monitoring in recent years, but most of them use linear features and do not take into account the complex nonlinearities of industrial processes. Therefore, a fault detection scheme based on temporal-spatial neighborhood enhanced sparse autoencoder is proposed in this paper. Firstly, it selects the temporal neighborhood and spatial neighborhood of the sample at the current time within the time window with a certain length, the spatial similarity and time serial correlation are used for weighted reconstruction, and the reconstruction combines the current sample as the input of the sparse stack autoencoder (SSAE) to extract the correlation features between the current sample and the neighborhood information. Two statistics are constructed for fault detection. Considering that both types of neighborhood information contain spatial-temporal structural features, Bayesian fusion strategy is used to integrate the two parts of the detection results. Finally, the superiority of the method in this paper is illustrated by a numerical example and the Tennessee Eastman process.

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Yamada, Toshihiro, and Kenta Yamamoto. "A second-order weak approximation of SDEs using a Markov chain without Lévy area simulation." Monte Carlo Methods and Applications 24, no. 4 (December 1, 2018): 289–308. http://dx.doi.org/10.1515/mcma-2018-2024.

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Abstract This paper proposes a new Markov chain approach to second-order weak approximations of stochastic differential equations (SDEs) driven by d-dimensional Brownian motion. The scheme is explicitly constructed by polynomials of Brownian motions up to second order, and any discrete moment-matched random variables or the Lévy area simulation method are not used. The required number of random variables is still d in one-step simulation of the implementation of the scheme. In the Markov chain, a correction term with Lie bracket of vector fields associated with SDEs appears as the cost of not using moment-matched random variables.

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Wang, Zhenyu, Qiang Ma, and Xiaohua Ding. "Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods." Mathematics 8, no. 12 (December 9, 2020): 2195. http://dx.doi.org/10.3390/math8122195.

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Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.

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Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan, Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan. "Numerical Spline Method for Simulation of Stochastic Differential Equations systems: طريقة شرائحية عددية لمحاكاة حل نظم من المعادلات التفاضلية العشوائية." Journal of natural sciences, life and applied sciences 5, no. 4 (December 27, 2021): 130–11. http://dx.doi.org/10.26389/ajsrp.l030621.

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In this paper, numerical spline method is presented with collocation two parameters for solving systems of multi-dimensional stochastic differential equations (SDEs). Multi-Wiener's time-continuous process is simulated as a discrete process, and then the mean-square stability of proposed method when applied to a system of two-dimensional linear SDEs is studied. The study shows that the method is mean-square stability and third-order convergent when applied to a system of linear and nonlinear SDEs. Moreover, the effectiveness of our method was tested by solving two test linear and non-linear problems. The numerical results show that the accuracy and applicability of the proposed method are worthy of attention.

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Wang, Jianmin, Chengfeng Zhu, Ziqiang Xiao, Qijun Zhao, and Junzhe Liu. "Numerical Analysis on the Bending Performance of Prestressed Superposing-Poured Composite Beams." Advances in Civil Engineering 2020 (August 29, 2020): 1–10. http://dx.doi.org/10.1155/2020/8897621.

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Aiming at the bending performance of the prestressed superposing-poured concrete beam, the numerical simulation on the composite beams poured with the normal weight concrete (NWC) superposed on the fibred ceramsite lightweight aggregate concrete (LWAC) was conducted. Three kinds of prestressing schemes, straight linear prestressing force, curved prestressing force not across the casting interface, and curved prestressing force across the casting interface, were simulated for comparison, and the influence of the casting interval time was also considered. Results indicate that the stiffness of the superposing-poured beam can be effectively strengthened by considered schemes of the prestressing force; in addition, there are certain increases on the ultimate load except imposing the straight linear prestressing force. As the curved prestressing force is imposed across the casting interface, the maximal interlayer slip of the casting interfacial transition zone (C-ITZ) approximately equals to that without the prestressing force. The scalar stiffness degradation (SDEG) of the C-ITZ for the casting interval time being 14 days is obvious because of the weakening on the bonding performance of the C-ITZ. Comparatively, the SDEG variation of the C-ITZ in the model with the curved prestressing force across the casting interface is smoother and smaller on the whole than the other two prestressed schemes for the case of the casting interval time being 14 days.

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43

Avikainen, Rainer. "On irregular functionals of SDEs and the Euler scheme." Finance and Stochastics 13, no. 3 (July 11, 2009): 381–401. http://dx.doi.org/10.1007/s00780-009-0099-7.

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Ma, Jin, Jiongmin Yong, and Yanhong Zhao. "Four step scheme for general Markovian forward-backward SDES." Journal of Systems Science and Complexity 23, no. 3 (June 2010): 546–71. http://dx.doi.org/10.1007/s11424-010-0145-8.

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Leobacher, Gunther, and Michaela Szölgyenyi. "A numerical method for SDEs with discontinuous drift." BIT Numerical Mathematics 56, no. 1 (February 21, 2015): 151–62. http://dx.doi.org/10.1007/s10543-015-0549-x.

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46

Ankirchner, Stefan, Thomas Kruse, and Mikhail Urusov. "Numerical approximation of irregular SDEs via Skorokhod embeddings." Journal of Mathematical Analysis and Applications 440, no. 2 (August 2016): 692–715. http://dx.doi.org/10.1016/j.jmaa.2016.03.055.

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47

Zähle, Henryk. "Weak Approximation of SDEs by Discrete-Time Processes." Journal of Applied Mathematics and Stochastic Analysis 2008 (March 23, 2008): 1–15. http://dx.doi.org/10.1155/2008/275747.

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We consider the martingale problem related to the solution of an SDE on the line. It is shown that the solution of this martingale problem can be approximated by solutions of the corresponding time-discrete martingale problems under some conditions. This criterion is especially expedient for establishing the convergence of population processes to SDEs. We also show that the criterion yields a weak Euler scheme approximation of SDEs under fairly weak assumptions on the driving force of the approximating processes.

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Yu, Hui, and Minghui Song. "Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients." Journal of Applied Mathematics 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/675781.

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The numerical methods in the current known literature require the stochastic differential equations (SDEs) driven by Poisson random measure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler's method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. The main aim is to investigate the convergence of the Euler method in probability to such equations with non-Lipschitz coefficients. Numerical example is given to demonstrate our results.

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Xie, Hongling. "An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables." AIP Advances 12, no. 7 (July 1, 2022): 075306. http://dx.doi.org/10.1063/5.0096285.

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There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

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Lamba, H., J. C. Mattingly, and A. M. Stuart. "An adaptive Euler-Maruyama scheme for SDEs: convergence and stability." IMA Journal of Numerical Analysis 27, no. 3 (November 2, 2006): 479–506. http://dx.doi.org/10.1093/imanum/drl032.

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