On a generalization of the stochastic integral
On a generalization of the stochastic integral
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DOI：
10.1137/1120030
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发表时间：
1976-03
期刊：
影响因子：
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通讯作者：
A. Skorokhod
中科院分区：
文献类型：
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作者：
A. Skorokhod
Wiener提出的关于Wiener过程的随机积分（例如见[1]）已在几个方向上推广。H. Cram 6 r [2]发展了关于正交随机测度的随机积分的一般理论（例如，可以在[3]，Chap. 4，4）。K. [It 6 l-4]推广了随机函数[（t）]的Wiener积分f（t）dw（t）的定义，使得当s<= t时，f（s）与增量w（u）-w（t）无关，当u>和f2（t）dt<00时.文[5]构造了非随机函数的多重Wiener积分。本文给出了关于正交高斯测度的积分的一个推广，类似于H。Cram 6 r积分，但现在适用于随机可积函数。我们注意到，在构造It 6积分时，定义相对于维纳过程构造的随机测度的空间的一维性以及与之相关的“过去”和“未来”的概念发挥了重要作用。一维性（积分是为独立于“未来”的函数定义的）。试图将It 6的构造转移到多个维度的情况下，遇到了以自然方式引入秩序的不可能性。下面研究的结构允许我们定义积分ff（x）tx（dx）
The stochastic integral with respect to a Wiener process, proposed by Wiener,(see, for example,[1]) has been generalized in several directions. H. Cram6r [2] developed a general theory of stochastic integrals with respect to orthogonal random measures (an account of this theory can be found, for example, in [3], Chap. 4, 4). K. It6 l-4] extended the definition of the Wiener integral f (t) dw (t) for random functions [(t), such that, for s<= t, f (s) is independent of the increments w (u)-w (t) for u> and f2 (t) dt< 00.Finally, K. It6 [5] constructed multiple Wiener integrals of non-random functions. Inthe present paper we propose a generalization ofan integral with respect to a Gaussian measure with orthogonal values, analogous to H. Cram6r’s integral, but now applicable to random integrable functions. We note that inconstructing the It6 integral, an essential role was played by the one-dimensionality of the space on which the random measure, constructed with respect to the Wiener process, was defined and the notions of" past" and" future" which are connected with this one-dimensionality (the integral was defined for functions independent of the" future"). Attempts to transfer It6’s construction to the case of several dimensions met with the impossibility of introducing order in a natural way. The construction investigated below permits usto define the integral ff (x) tx (dx)