Algebraic structure of stochastic expansions and efficient simulation

Ebrahimi‐Fard, Kurusch; Lundervold, Alexander Selvikvåg; Malham, Simon J. A.; Munthe-Kaas, Hans; Wiese, Anke

doi:10.1098/rspa.2012.0024

Cited by 16 publications

(29 citation statements)

References 30 publications

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“…On the other hand there is much literature relating the shuffle and quasishuffle products to stochastic integration, see e.g. [18]. We begin with the Stratonovich/shuffle case.…”

Section: The Shuffle and Quasishuffle Productsmentioning

confidence: 99%

Word combinatorics for stochastic differential equations: Splitting integrators

Alamo¹,

Sanz‐Serna²

2019

Communications on Pure &Amp; Applied Analysis

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We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local error; this makes it possible to easily formulate the conditions that guarantee that a given integrator achieves a prescribed strong or weak order. This approach bypasses the need to use the Baker-Campbell-Hausdorff (BCH) formula and shows the existence of an order barrier of two for the attainable weak order. The paper also provides a succinct introduction to the combinatorics of words.Mathematical Subject Classification (2010) 65C30, 60H05, 16T05

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“…On the other hand there is much literature relating the shuffle and quasishuffle products to stochastic integration, see e.g. [18]. We begin with the Stratonovich/shuffle case.…”

Section: The Shuffle and Quasishuffle Productsmentioning

confidence: 99%

Word combinatorics for stochastic differential equations: Splitting integrators

Alamo¹,

Sanz‐Serna²

2019

Communications on Pure &Amp; Applied Analysis

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“…In any case this scheme can be seen as an alternative to the established perturbative approach employed in quantum mechanical systems and quantum field theories (see also [13]), therefore algebraic and numerical techniques developed in solving SDEs will be of great relevance (see e.g. [6,28] and references therein). The significant observation is the existence of the heat kernel in front of dM 0 .…”

Section: Dynamical Diffusion Matrixmentioning

confidence: 99%

“…generalizing the notion of the Darboux-dressing transform) in the case of SPDEs will lead to a modified scheme for producing and solving certain types of SDEs/SPDEs [34]. Also, the study of the possible connections between the algebraic structures arising when solving SDEs [28], and the deformed algebras associated to integrable systems is a particularly interesting direction to pursue.…”

Section: Generalizations and Commentsmentioning

confidence: 99%

Stochastic analysis & discrete quantum systems

2019

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We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift plays the role of a super potential in the usual super-symmetric quantum mechanics sense. Some simple illustrative examples such as the Ornstein-Uhlenbeck process and the multidimensional Black-Scholes model are also discussed. Basic examples of quantum integrable systems such as the quantum discrete non-linear hierarchy (DNLS) and the XXZ spin chain are presented providing specific connections between quantum (integrable) systems and stochastic differential equations (SDEs). The continuum limits of the SDEs for the first two members of the NLS hierarchy turn out to be the stochastic transport and the stochastic heat equations respectively. The quantum Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are derived.

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“…. , µ, µ ∈ (1/2)N, (17) ensure that the series in (15) only comprises terms of size O(h µ+1/2 ). In fact, under suitable assumptions on (13), the fulfillment of the order conditions ensures that the local error possesses an O(h µ+1/2 ) bound (see the Appendix).…”

Section: The Local Errormentioning

confidence: 99%

A Technique for Studying Strong and Weak Local Errors of Splitting Stochastic Integrators

Alamo¹,

Sanz‐Serna²

2016

SIAM J. Numer. Anal.

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We present a technique, based on so-called word series, to write down in a systematic way expansions of the strong and weak local errors of splitting algorithms for the integration of Stratonovich stochastic differential equations. Those expansions immediately lead to the corresponding order conditions. Word series are similar to, but simpler than, the B-series used to analyze Runge-Kutta and other one-step integrators. The suggested approach makes it unnecessary to use the Baker-Campbell-Hausdorff formula. As an application, we compare two splitting algorithms recently considered by Leimkuhler and Matthews to integrate the Langevin equations. The word series method bears out clearly reasons for the advantages of one algorithm over the other.

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Algebraic structure of stochastic expansions and efficient simulation

Cited by 16 publications

References 30 publications

Word combinatorics for stochastic differential equations: Splitting integrators

Word combinatorics for stochastic differential equations: Splitting integrators

Stochastic analysis & discrete quantum systems

A Technique for Studying Strong and Weak Local Errors of Splitting Stochastic Integrators

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