lsamvy过程与拟洗牌代数

Pub Date : 2013-11-07 DOI:10.1080/17442508.2013.865131
Charles Curry, K. Ebrahimi-Fard, S. Malham, Anke Wiese
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引用次数: 11

摘要

研究了半鞅的重复积分代数。证明了半鞅的极小族生成拟洗牌代数。本质上,为了满足最小准则,首先,族必须是由它的元素和由族的元素递归构造的二次共变过程生成的重复积分代数的最小生成器。其次,递归构造的二次共变过程可能位于先前构造的二次共变过程和族的线性张成空间中,但可能不位于这些过程的重复积分的线性张成空间中。我们证明了具有有限矩的独立lsamvy过程的有限族产生极小族。证明的关键是Teugels鞅及其强正交化。我们得出一个有限族的独立lsamvy过程形成一个拟洗牌代数。我们讨论了构建由lsamvy过程驱动的随机微分方程强逼近的有效数值方法的重要潜在应用。
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Lévy processes and quasi-shuffle algebras
We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.
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