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On the Differentiability of Measures Which Correspond to Stochastic Processes. I. Processes with Independent Increments

A. V. Skorokhod

Kiev

Abstract: Kolmogorov (see [2] pg. 39) has proved that for each stochastic process there exists a corresponding unique measure on the minimal Borel field containing all cylindrical sets of the space of all functions.
Let $\xi_1(t)$ and $\xi_2(t)$ be processes with independent increments and $\mu_1$ and $\mu_2$ – measures corresponding to these processes. In this paper the conditions for which the measure $\mu_2$ is absolutely continuous with respect to the measure $\mu_1$ are investigated (Theorem A), and the density of the measure $\mu_2$ with respect to the measure $\mu_2$ is calculated (Theorem B).

Received: 14.04.1957

 English version: Theory of Probability and its Applications, 1957, 2:4, 407–432