An Application of the Stratification Method to the Study of Functionals of Processes with Independent Increments M.A. Lifshits Let X = (Xt , 0 ≤ t ≤ 1) be a stochastically continuous process with independent increments. We consider the distribution of a stochastic functional Φ(X) and give sufficient conditions under which it is absolutely continuous with respect to Lebesgue measure or even has a bounded density. Let D = D[0, 1] denote standard Skorokhod space. Theorem 1 Let the control measure Π of X be absolutely continuous with respect to Lebesgue measure on R1 × [0, 1] and for every distinct t1 , t2 ∈ [0, 1] Z

∞

Z

t2

Π(dx, dt) = +∞. −∞

t1

Let Φ : D → R1 be a functional which satisfies the following assumption of partial derivability: For some S ⊂ D one has P (X ∈ S) = 1 and the partial dervative Φ0· (·) is defined on D × S. Moreover, for every x ∈ S there exists a direction l(x) ∈ D such that Φ0l(x) (x) 6= 0 and the function Φ0· (·) is continuous at (l(x), x) with respect to Skorokhod topology. Then the distribution of r.v. Φ(X) is absolutely continuous. The typical example to which this theorem can be applied, is an integral functional Z 1 Φ(x) = f (x(t))dt, 0

provided that f is a locally Lipschitz function, and that f 0 exists and is continuous and non-zero on a set of full Lebesgue measure in R1 . We also investigate the boundedness of the density of the functional Ψ(X) assuming that the process X is α-stable and Z 1 Ψ(x) = g(x(t), t)dt. 0

The main assumption here is a (very weak) exponential lower bound on |gx0 (x, t)| related to the so called small ball behavior of X.

1