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Academic literature on the topic 'Numerical scheme for SDEs'

Author: Grafiati

Published: 9 September 2022

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Journal articles on the topic "Numerical scheme for SDEs"

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 (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 (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 (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 (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 (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 (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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