An Algebra of Non-commutative Bounded Semimartingales: Square and Angle Quantum Brackets
An Algebra of Non-commutative Bounded Semimartingales: Square and Angle Quantum Brackets
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非交换有界半鞅代数：方量子括号和角量子括号
DOI：
10.1006/jfan.1994.1109
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发表时间：
1994
影响因子：
1.7
通讯作者：
S. Attal
中科院分区：
文献类型：
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作者：
S. Attal
摘要利用Attal和Meyer在Fock空间上对非交换随机微积分的推广，给出了非交换半鞅复合的一个* -代数。我们根据半鞅对某些Radon测度的正则性给出了这个代数的一个表征。这种表征是对Parthasarathy和Sinha的量子鞅表示定理的量子半鞅的推广[J. Funct]。中华医学杂志。67(1986),126-151。我们通过定义非交换方括号和角括号，发展了非交换随机微积分，它们是经典的扩展，并验证了大多数常用性质。得到了非交换半鞅多项式的一个非交换伊藤公式（也对应于交换情况下的一般伊藤公式）。通过证明方括号可以被解释为过程的二次变分，给出了方括号的内在定义。最后，给出了具有任意（最多可数）多重的Fock空间的所有结果。
Abstract Thanks to the extension of the non-commutative stochastic calculus on Fock space developed by Attal and Meyer, we give a ∗-algebra for the composition of non-commutative semimartingales. We give a characterization of this algebra in terms of the regularity of the semimartingales with respect to some Radon measures. This characterization is an extension for quantum semimartingales of the quantum martingale representation theorem of Parthasarathy and Sinha [ J. Funct. Anal. 67 (1986), 126-151]. We develop a non-commutative stochastic calculus by defining non-commutative square and angle brackets, which are extensions of the classical ones and verify most of the usual properties. A non-commutative Ito formula for polynomials of non-commutative semimartingales (which also corresponds to the usual one in the commutative case) is obtained. An intrinsic definition of the square bracket is given by proving that it can be interpreted as a quadratic variation of the processes. Finally, all the corresponding results for a Fock space with any, at most countable, multiplicity are given.