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Автор: Grafiati
Опубліковано: 10 грудня 2022
Оновлено: 29 січня 2023

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1

Pieschner, Susanne, and Christiane Fuchs. "Bayesian inference for diffusion processes: using higher-order approximations for transition densities." Royal Society Open Science 7, no. 10 (October 2020): 200270. http://dx.doi.org/10.1098/rsos.200270.

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Анотація:
Modelling random dynamical systems in continuous time, diffusion processes are a powerful tool in many areas of science. Model parameters can be estimated from time-discretely observed processes using Markov chain Monte Carlo (MCMC) methods that introduce auxiliary data. These methods typically approximate the transition densities of the process numerically, both for calculating the posterior densities and proposing auxiliary data. Here, the Euler–Maruyama scheme is the standard approximation technique. However, the MCMC method is computationally expensive. Using higher-order approximations may accelerate it, but the specific implementation and benefit remain unclear. Hence, we investigate the utilization and usefulness of higher-order approximations in the example of the Milstein scheme. Our study demonstrates that the MCMC methods based on the Milstein approximation yield good estimation results. However, they are computationally more expensive and can be applied to multidimensional processes only with impractical restrictions. Moreover, the combination of the Milstein approximation and the well-known modified bridge proposal introduces additional numerical challenges.
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2

Sengul, Suleyman, Zafer Bekiryazici, and Mehmet Merdan. "Wong-Zakai method for stochastic differential equations in engineering." Thermal Science 25, Spec. issue 1 (2021): 131–42. http://dx.doi.org/10.2298/tsci200528014s.

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Анотація:
In this paper, Wong-Zakai approximation methods are presented for some stochastic differential equations in engineering sciences. Wong-Zakai approximate solutions of the equations are analyzed and the numerical results are compared with results from popular approximation schemes for stochastic differential equations such as Euler-Maruyama and Milstein methods. Several differential equations from engineering problems containing stochastic noise are investigated as numerical examples. Results show that Wong-Zakai method is a reliable tool for studying stochastic differential equations and can be used as an alternative for the known approximation techniques for stochastic models.
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3

Azadfar, Hamed, and Parisa Nabati. "New truncated Milstein approximation of solution of stochastic differential equations." Communications on Advanced Computational Science with Applications 2018, no. 1 (2018): 15–25. http://dx.doi.org/10.5899/2018/cacsa-00090.

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4

Park, Hail. "Estimation of affine term structure models under the Milstein approximation." Applied Economics Letters 21, no. 9 (March 5, 2014): 651–56. http://dx.doi.org/10.1080/13504851.2014.881962.

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5

Ghayebi, B., and S. M. Hosseini. "A Simplified Milstein Scheme for SPDEs with Multiplicative Noise." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/140849.

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Анотація:
This paper deals with a research question raised by Jentzen and Röckner (A Milstein scheme for SPDEs, arXiv:1001.2751v4 (2012)), whether the exponential term in their introduced scheme can be replaced by a simpler mollifier. This replacement can lead to more simplification and computational reduction in simulation. So, in this paper, we essentially replace the exponential term with a Padé approximation of order 1 and denote the resulting scheme by simplified Milstein scheme. The convergence analysis for this scheme is carried out and it is shown that even with this replacement the order of convergence is maintained, while the resulting scheme is easier to implement and slightly more efficient computationally. Some numerical tests are given that confirm the order of accuracy and also computational cost reduction.
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6

Ranjbar, Hassan, Leila Torkzadeh, Dumitru Baleanu, and Kazem Nouri. "Simulating systems of Itô SDEs with split-step $ (\alpha, \beta) $-Milstein scheme." AIMS Mathematics 8, no. 2 (2022): 2576–90. http://dx.doi.org/10.3934/math.2023133.

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Анотація:
<abstract><p>In the present study, we provide a new approximation scheme for solving stochastic differential equations based on the explicit Milstein scheme. Under sufficient conditions, we prove that the split-step $ (\alpha, \beta) $-Milstein scheme strongly convergence to the exact solution with order $ 1.0 $ in mean-square sense. The mean-square stability of our scheme for a linear stochastic differential equation with single and multiplicative commutative noise terms is studied. Stability analysis shows that the mean-square stability of our proposed scheme contains the mean-square stability region of the linear scalar test equation for suitable values of parameters $ \alpha, \beta $. Finally, numerical examples illustrate the effectiveness of the theoretical results.</p></abstract>
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7

Slassi, Mehdi. "A Milstein-based free knot spline approximation for stochastic differential equations." Journal of Complexity 28, no. 1 (February 2012): 37–47. http://dx.doi.org/10.1016/j.jco.2011.03.005.

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8

Mrázek, Milan, and Jan Pospíšil. "Calibration and simulation of Heston model." Open Mathematics 15, no. 1 (May 23, 2017): 679–704. http://dx.doi.org/10.1515/math-2017-0058.

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Анотація:
Abstract We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods. We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.
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9

Koulis, Theodoro, Alexander Paseka, and Aerambamoorthy Thavaneswaran. "Recursive Estimation for Continuous Time Stochastic Volatility Models Using the Milstein Approximation." Journal of Mathematical Finance 03, no. 03 (2013): 357–65. http://dx.doi.org/10.4236/jmf.2013.33036.

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10

Barth, Andrea, and Annika Lang. "Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises." Applied Mathematics & Optimization 66, no. 3 (August 10, 2012): 387–413. http://dx.doi.org/10.1007/s00245-012-9176-y.

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11

Kamrani, Minoo, and Nahid Jamshidi. "Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation." Numerical Algorithms 79, no. 2 (November 13, 2017): 357–74. http://dx.doi.org/10.1007/s11075-017-0440-8.

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12

Brault, Antoine, and Antoine Lejay. "The non-linear sewing lemma III: Stability and generic properties." Forum Mathematicum 32, no. 5 (September 1, 2020): 1177–97. http://dx.doi.org/10.1515/forum-2019-0309.

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Анотація:
AbstractSolutions of Rough Differential Equations (RDE) may be defined as paths whose increments are close to an approximation of the associated flow. They are constructed through a discrete scheme using a non-linear sewing lemma. In this article, we show that such solutions also solve a fixed point problem by exhibiting a suitable functional. Convergence then follows from consistency and stability, two notions that are adapted to our framework. In addition, we show that uniqueness and convergence of discrete approximations is a generic property, meaning that it holds excepted for a set of vector fields and starting points which is of Baire first category. At last, we show that Brownian flows are almost surely unique solutions to RDE associated to Lipschitz flows. The later property yields almost sure convergence of Milstein schemes.
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13

Calzolari, Antonella, Patrick Florchinger, and Giovanna Nappo. "Nonlinear filtering for stochastic systems with fixed delay: Approximation by a modified Milstein scheme." Computers & Mathematics with Applications 61, no. 9 (May 2011): 2498–509. http://dx.doi.org/10.1016/j.camwa.2011.02.036.

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14

Hofmann, Norbert, and Thomas Müller-Gronbach. "A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag." Journal of Computational and Applied Mathematics 197, no. 1 (December 2006): 89–121. http://dx.doi.org/10.1016/j.cam.2005.10.027.

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15

Morkisz, Paweł M., and Paweł Przybyłowicz. "Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information." Journal of Computational and Applied Mathematics 383 (February 2021): 113112. http://dx.doi.org/10.1016/j.cam.2020.113112.

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16

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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Анотація:
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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17

Jacob Kayode, Sunday. "Effect of Varying StepSizes in Numerical Approximation of Stochastic Differential Equations Using One Step Milstein Method." Applied and Computational Mathematics 4, no. 5 (2015): 351. http://dx.doi.org/10.11648/j.acm.20150405.14.

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18

Lay, Harold A., Zane Colgin, Viktor Reshniak, and Abdul Q. M. Khaliq. "On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters." Monte Carlo Methods and Applications 24, no. 4 (December 1, 2018): 309–21. http://dx.doi.org/10.1515/mcma-2018-2025.

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Abstract We explore different methods of solving systems of stochastic differential equations by first implementing the Euler–Maruyama and Milstein methods with a Monte Carlo simulation on a CPU. The performance of the methods is significantly improved through the recently developed antithetic multilevel Monte Carlo estimator, which yields a computation complexity of {\mathcal{O}(\epsilon^{-2})} root-mean-square error and does so without the approximation of Lévy areas. Further improvements in performance are gained by moving the algorithms to a GPU - first on a single device and then on a multi-GPU cluster. Our GPU implementation of the antithetic multilevel Monte Carlo displays a major speedup in computation when compared with many commonly used approaches in the literature. While our work is focused on the simulation of the stochastic volatility and interest rate model, it is easily extendable to other stochastic systems, and it is of particular interest to those with non-diagonal, non-commutative noise.
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19

Ilie, Silvana, and Monjur Morshed. "Adaptive Time-Stepping Using Control Theory for the Chemical Langevin Equation." Journal of Applied Mathematics 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/567275.

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Stochastic modeling of biochemical systems has been the subject of intense research in recent years due to the large number of important applications of these systems. A critical stochastic model of well-stirred biochemical systems in the regime of relatively large molecular numbers, far from the thermodynamic limit, is the chemical Langevin equation. This model is represented as a system of stochastic differential equations, with multiplicative and noncommutative noise. Often biochemical systems in applications evolve on multiple time-scales; examples include slow transcription and fast dimerization reactions. The existence of multiple time-scales leads to mathematical stiffness, which is a major challenge for the numerical simulation. Consequently, there is a demand for efficient and accurate numerical methods to approximate the solution of these models. In this paper, we design an adaptive time-stepping method, based on control theory, for the numerical solution of the chemical Langevin equation. The underlying approximation method is the Milstein scheme. The adaptive strategy is tested on several models of interest and is shown to have improved efficiency and accuracy compared with the existing variable and constant-step methods.
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20

Yan, Feng, Salah-Eldin A. Mohammed, and Yaozhong Hu. "Discrete-time approximations of stochastic delay equations: The Milstein scheme." Annals of Probability 32, no. 1A (January 2004): 265–314. http://dx.doi.org/10.1214/aop/1078415836.

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21

Ermakov, Sergej M., and Anna A. Pogosian. "On solving stochastic differential equations." Monte Carlo Methods and Applications 25, no. 2 (June 1, 2019): 155–61. http://dx.doi.org/10.1515/mcma-2019-2038.

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Abstract This paper proposes a new approach to solving Ito stochastic differential equations. It is based on the well-known Monte Carlo methods for solving integral equations (Neumann–Ulam scheme, Markov chain Monte Carlo). The estimates of the solution for a wide class of equations do not have a bias, which distinguishes them from estimates based on difference approximations (Euler, Milstein methods, etc.).
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22

Koulis, Theodoro, and Aera Thavaneswaran. "Inference for Interest Rate Models Using Milstein’s Approximation." Journal of Mathematical Finance 03, no. 01 (2013): 110–18. http://dx.doi.org/10.4236/jmf.2013.31010.

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23

Kumar, Chaman, and Sotirios Sabanis. "On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients." BIT Numerical Mathematics 59, no. 4 (June 19, 2019): 929–68. http://dx.doi.org/10.1007/s10543-019-00756-5.

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24

Kumar, Chaman, and Sotirios Sabanis. "Correction to: On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients." BIT Numerical Mathematics 60, no. 2 (October 8, 2019): 537. http://dx.doi.org/10.1007/s10543-019-00780-5.

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25

von Hallern, Claudine, and Andreas Rößler. "A derivative-free Milstein type approximation method for SPDEs covering the non-commutative noise case." Stochastics and Partial Differential Equations: Analysis and Computations, October 4, 2022. http://dx.doi.org/10.1007/s40072-022-00274-6.

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Анотація:
AbstractWe propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need to fulfill a commutativity condition for the noise term. The newly developed derivative-free Milstein type scheme differs significantly from schemes that are appropriate for the case of commutative noise. As a key result, the new derivative-free Milstein type scheme needs only two stages that are specifically tailored based on a technique that, compared to the original Milstein scheme, allows for a reduction of the computational complexity by one order of magnitude. Moreover, the proposed derivative-free Milstein scheme can flexibly be combined with some approximation method for the involved iterated stochastic integrals. As the main result, we prove the strong $$L^2$$ L 2 -convergence of the introduced derivative-free Milstein type scheme, especially if it is combined with any suitable approximation algorithm for the necessary iterated stochastic integrals. We carry out a rigorous analysis of the error versus computational cost and derive the effective order of convergence for the derivative-free Milstein type scheme in the case that the truncated Fourier series algorithm for the approximation of the iterated stochastic integrals is applied. As a further novelty, we show that the use of approximations of iterated stochastic integrals based on truncated Fourier series together with the proposed derivative-free Milstein type scheme improves the effective order of convergence compared to that of the Euler scheme and the original Milstein scheme. This result is contrary to well known results in the finite dimensional SDE case where the use of merely truncated Fourier series does not improve the effective order of convergence in the $$L^2$$ L 2 -sense compared to that of the Euler scheme.
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26

Luong, Duc-Trong, and Hoang-Long Ngo. "Semi-implicit Milstein approximation scheme for non-colliding particle systems." Calcolo 56, no. 3 (July 9, 2019). http://dx.doi.org/10.1007/s10092-019-0319-2.

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27

Davie, A. M. "Approximation of SDE solutions using local asymptotic expansions." Stochastics and Partial Differential Equations: Analysis and Computations, December 10, 2021. http://dx.doi.org/10.1007/s40072-021-00232-8.

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AbstractWe develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.
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28

Şengül, Süleyman, and Mehmet Merdan. "Wong-Zakai Method Applications for Explicitly Solvable Stochastic Differential Equations." Journal of Advances in Mathematics and Computer Science, October 24, 2019, 1–12. http://dx.doi.org/10.9734/jamcs/2019/v34i1-230202.

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Анотація:
In this study, three Ito stochastic differential equations with multiplicative noise are investigated with Wong-Zakai method. The stochastic differential equations are also analyzed by Euler-Maruyama, Milstein and Runge Kutta stochastic approximation methods. The relative errors of these three methods are compared and the performance of Wong-Zakai method is shown alongside numerical results.
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29

van Rhijn, Jorino, Cornelis W. Oosterlee, Lech A. Grzelak, and Shuaiqiang Liu. "Monte Carlo simulation of SDEs using GANs." Japan Journal of Industrial and Applied Mathematics, September 23, 2022. http://dx.doi.org/10.1007/s13160-022-00534-x.

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AbstractGenerative adversarial networks (GANs) have shown promising results when applied on partial differential equations and financial time series generation. We investigate if GANs can also be used to approximate one-dimensional It$$\hat{\text {o}}$$ o ^ stochastic differential equations (SDEs). We propose a scheme that approximates the path-wise conditional distribution of SDEs for large time steps. Standard GANs are only able to approximate processes in distribution, yielding a weak approximation to the SDE. A conditional GAN architecture is proposed that enables strong approximation. We inform the discriminator of this GAN with the map between the prior input to the generator and the corresponding output samples, i.e. we introduce a ‘supervised GAN’. We compare the input-output map obtained with the standard GAN and supervised GAN and show experimentally that the standard GAN may fail to provide a path-wise approximation. The GAN is trained on a dataset obtained with exact simulation. The architecture was tested on geometric Brownian motion (GBM) and the Cox–Ingersoll–Ross (CIR) process. The supervised GAN outperformed the Euler and Milstein schemes in strong error on a discretisation with large time steps. It also outperformed the standard conditional GAN when approximating the conditional distribution. We also demonstrate how standard GANs may give rise to non-parsimonious input-output maps that are sensitive to perturbations, which motivates the need for constraints and regularisation on GAN generators.
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30

Kubilius, Kęstutis, and Dmitrij Melichov. "Estimating the Hurst index of the solution of a stochastic integral equation." Lietuvos matematikos rinkinys 50 (December 20, 2009). http://dx.doi.org/10.15388/lmr.2009.04.

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Let X(t) be a solution of a stochastic integral equation driven by fractional Brownian motion BH and let V2n (X, 2) = \sumn-1 k=1(\delta k2X)2 be the second order quadratic variation, where \delta k2X = X (k+1/N) − 2X (k/ n) +X (k−1/n). Conditions under which n2H−1Vn2(X, 2) converges almost surely as n → ∞ was obtained. This fact is used to get a strongly consistent estimator of the Hurst index H, 1/2 < H < 1. Also we show that this estimator retains its properties if we replace Vn2(X, 2) with Vn2(Y, 2), where Y (t) is the Milstein approximation of X(t).
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