ABSTRACT

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 generalization. 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 necessarily 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.

摘要

我们观察到在一些抽象 Banach 空间B上定义的无限维独立增量过程的条件期望的多线性保留特性。它本质上类似于为有限维随机过程大量分析的多项式保持特性，因此提供了无限维的概括。然而，虽然多项式是使用乘法运算符定义的，因此需要巴拿赫代数结构，但我们在此证明的多重线性保持性质即使对于在不一定是巴拿赫代数的巴拿赫空间上定义的过程也成立。在B的特殊情况下作为可交换的 Banach 代数，我们表明独立增量过程是多项式过程，在某种意义上与有限维情况下多项式过程的规范扩展相吻合。交换性的假设被证明是至关重要的，并且在非交换性 Banach 代数中，多重线性概念自然出现。我们的一些结果超越了独立的增量过程，因此一般地阐明了无限维多项式过程。