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.

我们观察到在某些抽象Banach空间B上定义的无穷维独立增量过程的条件期望的多线性保持性质。它在本质上与为有限维随机过程进行了大量分析的多项式保持性质相似，因此提供了无穷维概括。但是，尽管多项式是使用乘法运算符定义的，因此需要使用Banach代数结构，但即使对于在不一定是Banach代数的Banach空间上定义的过程，我们在此处证明的多线性保持性质也成立。在B的特殊情况下作为可交换的Banach代数，我们证明了独立的增量过程是多项式过程，从某种意义上讲，它与多项式过程在有限维情况下的典范扩展一致。可交换性的假设被证明是至关重要的，并且在非可交换Banach代数中，多线性概念自然而然地出现了。我们的某些结果超出了独立的增量过程的范围，因此通常可以说明无限维多项式过程。