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

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 ₁ (t) and ₂ (t) be processes with independent increments and ₁ and ₂ — measures corresponding to these processes. In this paper the conditions for which the measure ₂ is absolutely continuous with respect to the measure ₁ are investigated (Theorem A), and the density of the measure ₂ with respect to the measure ₂ is calculated (Theorem B).