Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps
Abstract We study the problem of approximating the expected value $${\mathbb E}f(X(1))$$ of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their...
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| Autor*in: |
Müller-Gronbach, Thomas [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015 |
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| Schlagwörter: |
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| Anmerkung: |
© SFoCM 2015 |
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| Übergeordnetes Werk: |
Enthalten in: Foundations of computational mathematics - Springer US, 2001, 16(2015), 5 vom: 20. Aug., Seite 1325-1366 |
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| Übergeordnetes Werk: |
volume:16 ; year:2015 ; number:5 ; day:20 ; month:08 ; pages:1325-1366 |
| Links: |
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| DOI / URN: |
10.1007/s10208-015-9277-5 |
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| Katalog-ID: |
OLC2057963004 |
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| 520 | |a Abstract We study the problem of approximating the expected value $${\mathbb E}f(X(1))$$ of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying a Markov transition based on the distribution of a simplified weak Itô–Taylor step together with strategies to reduce the diameter and the size of the support of a discrete measure. We essentially assume that the coefficients of the SDE are s-times continuously differentiable and the diffusion coefficient satisfies a uniform non-degeneracy condition and that the integrand f is r-times continuously differentiable. In the case $$r \le (\lfloor s/2 \rfloor - 1) \cdot 2d/(d + 2)$$, we almost achieve an error of order $$\min (r, s)/d$$ in terms of the computational cost, which is optimal in a worst-case sense. | ||
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Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps |
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Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps |
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Müller-Gronbach, Thomas |
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deterministic quadrature formulas for sdes based on simplified weak itô–taylor steps |
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Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps |
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Abstract We study the problem of approximating the expected value $${\mathbb E}f(X(1))$$ of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying a Markov transition based on the distribution of a simplified weak Itô–Taylor step together with strategies to reduce the diameter and the size of the support of a discrete measure. We essentially assume that the coefficients of the SDE are s-times continuously differentiable and the diffusion coefficient satisfies a uniform non-degeneracy condition and that the integrand f is r-times continuously differentiable. In the case $$r \le (\lfloor s/2 \rfloor - 1) \cdot 2d/(d + 2)$$, we almost achieve an error of order $$\min (r, s)/d$$ in terms of the computational cost, which is optimal in a worst-case sense. © SFoCM 2015 |
| abstractGer |
Abstract We study the problem of approximating the expected value $${\mathbb E}f(X(1))$$ of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying a Markov transition based on the distribution of a simplified weak Itô–Taylor step together with strategies to reduce the diameter and the size of the support of a discrete measure. We essentially assume that the coefficients of the SDE are s-times continuously differentiable and the diffusion coefficient satisfies a uniform non-degeneracy condition and that the integrand f is r-times continuously differentiable. In the case $$r \le (\lfloor s/2 \rfloor - 1) \cdot 2d/(d + 2)$$, we almost achieve an error of order $$\min (r, s)/d$$ in terms of the computational cost, which is optimal in a worst-case sense. © SFoCM 2015 |
| abstract_unstemmed |
Abstract We study the problem of approximating the expected value $${\mathbb E}f(X(1))$$ of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying a Markov transition based on the distribution of a simplified weak Itô–Taylor step together with strategies to reduce the diameter and the size of the support of a discrete measure. We essentially assume that the coefficients of the SDE are s-times continuously differentiable and the diffusion coefficient satisfies a uniform non-degeneracy condition and that the integrand f is r-times continuously differentiable. In the case $$r \le (\lfloor s/2 \rfloor - 1) \cdot 2d/(d + 2)$$, we almost achieve an error of order $$\min (r, s)/d$$ in terms of the computational cost, which is optimal in a worst-case sense. © SFoCM 2015 |
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Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps |
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