The algebra of iterated stochastic integrals

Gaines, J. G.

doi:10.1080/17442509408833918

Cited by 47 publications

(52 citation statements)

References 14 publications

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“…The use of Hopf shuffle algebras in stochastic expansions can be traced through, for example, Strichartz (1987), Reutenauer (1993), Gaines (1994), Li & Liu (2000), Kawski (2002), Baudoin (2004), Ebrahimi-Fard & Guo (2006), Murua (2006), Lyons et al (2007) and Manchon & Paycha (2007), to name a few. The paper by Munthe-Kaas & Wright (2008) on the Hopf algebraic of Lie group integrators actually instigated the Hopf algebra direction adopted here.…”

Section: The Linear Word-to-vector Field Mapmentioning

confidence: 99%

Stochastic expansions and Hopf algebras

Malham

Wiese

2009

Proc. R. Soc. A.

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We study solutions to nonlinear stochastic differential systems driven by a multidimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell-Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that, at low orders, this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However, it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here, we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof uses the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies.

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Section: The Linear Word-to-vector Field Mapmentioning

confidence: 99%

Stochastic expansions and Hopf algebras

Malham

Wiese

2009

Proc. R. Soc. A.

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“…Hence in both cases the quadrature effort is proportional to QN . Implicit in the relations (8.1) is the natural underlying shuffle algebra created by integration by parts (see Gaines [20,19], Kawksi [32] and Munthe-Kaas and Wright [48]). Two further results are of interest.…”

Section: Uniformly Accurate Magnus Integratorsmentioning

confidence: 99%

Efficient Strong Integrators for Linear Stochastic Systems

Lord¹,

Malham²,

Wiese³

2008

SIAM J. Numer. Anal.

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Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. Firstly, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Secondly, we derive the maximal rate of convergence for arbitrary multi-dimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with non-commuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Thirdly, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control.

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“…The order of a strong numerical method is determined by the set of multiple Wiener integrals simulated and included. Since the set of all multiple (Stratonovich) Wiener integrals is generated by those based on Lyndon words (see Reutenauer 1993, p. 111 andGaines 1994), we need only simulate the multiple Wiener integrals indexed by Lyndon words. The other multiple Wiener integrals of that order can be computed by linear combinations of products of the appropriate Lyndon word multiple integrals of that order or less.…”

Section: Introductionmentioning

confidence: 99%

Algebraic structure of stochastic expansions and efficient simulation

et al. 2012

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We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems. Our motivation is to construct efficient integrators. These are approximations that generate strong numerical integration schemes that are more accurate than the corresponding stochastic Taylor approximation, independent of the governing vector fields and to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is one example. Herein, we show that the natural context to study stochastic integrators and their properties is the convolution shuffle algebra of endomorphisms; establish a new whole class of efficient integrators; and then prove that, within this class, the sinhlog integrator generates the optimal efficient stochastic integrator at all orders.

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The algebra of iterated stochastic integrals

Cited by 47 publications

References 14 publications

Stochastic expansions and Hopf algebras

Stochastic expansions and Hopf algebras

Efficient Strong Integrators for Linear Stochastic Systems

Algebraic structure of stochastic expansions and efficient simulation

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