Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

Smith, William E.; Sloan, Ian H.; Opie, A H

doi:10.1090/s0025-5718-1983-0689468-1

Cited by 13 publications

(4 citation statements)

References 22 publications

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“…Analogously, as discussed in [26], for any function f ∈ F k 1 (R) whose growth at infinity satisfies the condition in Theorem 1 in [26], the error of the one-dimensional Gauss-Hermite rule is given by…”

Section: )mentioning

confidence: 94%

A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations

Zhang

Gunzburger²,

Zhang³

2013

JCM

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A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.Mathematics subject classification: 60H10, 60H35, 65C10, 65C20, 65C50.Because analytical solutions of BSDEs are often very difficult to obtain, approximate numerical solutions of BSDEs become highly desired in practical applications. There are mainly two types of numerical methods for BSDEs. One is based on the relation between the forwardbackward stochastic differential equations (FBSDEs) and corresponding parabolic partial differential equation (PDEs) [13,14,22]; the other is directly based on BSDEs or FBSDEs [2,3,6,7,10,12,23,27,28,30,31]. Zhao et al. proposed a θ-scheme for BSDEs in [28]; in [29], it was extended to a generalized θ-scheme. In [31], a stable multi-step scheme was proposed which is a highly accurate numerical method for BSDEs. Note that for the second type of numerical methods, approximating spatial derivatives at different time-space points for the case of PDEs is converted to approximating conditional mathematical expectations with Gaussian kernels centered at different time-space points.It should be noted that the BSDEs used in practice usually involve a multi-dimensional Brownian motion, such as the option pricing problem with multiple underlying assets. Existing numerical methods for BSDEs can be theoretically extended to the multi-dimensional cases; however, the computational cost may be unaffordable due to the so-called curse of dimensionality. The most popular approach to solving multi-dimensional BSDEs is the Monte Carlo method [3,28] that is very easy to implement. However, the convergence rate is typically very slow, although having a mild dependence on the dimensionality. Thus, an accurate and efficient numerical method for solving multi-dimensional BSDEs is highly desired in the BSDE community.In this paper, we extend the multi-step method in [31] using the sparse-grid method for solving multi-dimensional BSDEs. As discussed in [31], the target BSDE (1.1) is discretized by the multi-step scheme in the time direction. In the spatial domain, a quadrature rule is needed to approximate all the conditional mathematical expectations (multi-dimensional integrals) and an interpolation scheme is also needed to evaluate the integrands of the expectations at nongrid quadrature points. The sparse-grid method is highly suitable for the multi-step scheme because it has been demonstrated to be effective and efficient in dea...

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Section: )mentioning

confidence: 94%

A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations

Zhang

Gunzburger²,

Zhang³

2013

JCM

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“…where N Q is the number of spare grid quadrature points and the constant C q,k depends on q and the upper bound of the k-th derivative of f . Analogously for functions f ∈ F k 1 (R), we have the following error estimates in [25],…”

Section: Function Approximations On Sparse Gridsmentioning

confidence: 99%

Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

Fu¹,

Zhao²,

Zhou³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. We use the sparse grid Gaussian-Hermite quadrature rule to approximate the conditional expectations. And for the associated high dimensional interpolations, we adopt an spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and high accurate approximations in high-dimensions, provided that the solutions are sufficiently smooth. Several numerical examples are presented to demonstrate the efficiency of the proposed methods.

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“…Several authors, e.g., [5], [7], [9], [14], [16]- [19] have considered convergence of product integration rules over finite and infinite intervals, but generally their results have been given in terms of orders of convergence without particular attention to obtaining sharp error bounds. Several authors, e.g., [5], [7], [9], [14], [16]- [19] have considered convergence of product integration rules over finite and infinite intervals, but generally their results have been given in terms of orders of convergence without particular attention to obtaining sharp error bounds.…”

Section: Wnmentioning

confidence: 99%

Optimal Nodes for Interpolatory Product Integration

Smith¹,

Paget²

1992

SIAM J. Numer. Anal.

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This paper is concerned with choices of nodes in numerical integration rules to evaluate (kf).Here k is any element of a normed space corresponding to some integrability requirement, and f is many times differentiable, conditions which are appropriate to the practical application of such rules. By examining the behaviour of the error functional it is shown how optimal nodes can be defined in these circumstances.For finite intervals and a variety of integrability conditions on k, the resulting optimal nodes are defined and compared. Further results, although less complete, are obtained for similar integrals over the whole of and over +.

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Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

Cited by 13 publications

References 22 publications

A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations

A Sparse-Grid Method for Multi-Dimensional Backward Stochastic Differential Equations

Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

Optimal Nodes for Interpolatory Product Integration

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