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