独立增量过程：多重线性保持性质
Independent increment processes: A multilinearity preserving property
摘要：
我们观察到定义在抽象巴拿赫空间$B$上的无限维独立增量过程的条件期望的多重线性保持性质。它在性质上类似于多项式保持属性分析大大有限维随机过程，从而提供了一个无限维的推广。然而，虽然多项式的定义使用乘法运算符，因此需要一个巴拿赫代数结构，多重线性保持属性，我们证明在这里举行，即使是定义在一个巴拿赫空间，这是不必要的巴拿赫代数的过程。在特殊的情况下，$B$是一个交换的巴拿赫代数，我们表明，独立的增量过程是多项式过程的意义上，符合从有限维情况下的多项式过程的规范扩展。交换性的假设是至关重要的，在非交换巴拿赫代数的多重线性的概念自然产生。我们的一些结果持有超越独立的增量过程，从而揭示了无限维多项式过程一般。
英文摘要：
We observe a multilinearity preserving property of conditional expectation for infinite dimensional independent increment processes defined on some abstract Banach space $B$. It is similar in nature to the polynomial preserving property analysed greatly for finite dimensional stochastic processes and thus offers an infinite dimensional generalisation. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessary a Banach algebra. In the special case of $B$ being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite dimensional polynomial processes in general.
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