二阶矩过程为n阶均方可导的一个充要条件

A Necessary and Sufficient Condition of n-Order Mean Square Differentiability for a Second Moment Stochastic Process

一般的随机过程教材中只对一个二阶矩过程均方可导与其相关函数广义可导的等价性进行论述.将广义导数推广到n阶的情形,利用数学归纳法证明了二阶矩过程{X(t),t∈R}为n阶均方可导与其相关函数n 阶广义可导之间的等价性,并给出了判断二元函数f(s,t)为n阶广义可导的一个充分条件.

In general textbooks of stochastic process,only the equivalency between mean square differentiability of the process and generalized differentiability of the relevant function about a second moment stochastic process is expounded. In this paper, generalized derivative is extended to n-order case, and the equivalency between n-order mean square differentiability of the process and n-order generalized differentiability of relevant function is proved by means of mathematical induction. In addition, a sufficient condition judging n-order generalized differentiability for f(s,t) is given.