Stochastic Integrals and the Lévy–Ito Decomposition Theorem on Separable Banach Spaces

Abstract

A direct definition of stochastic integrals for deterministic Banach valued functions on separable Banach spaces with respect to compensated Poisson random measures is given. This definition yields a direct proof of the Lévy–Ito decomposition of a càdlàg process with stationary, independent increments into a jump and a Brownian component. It turns out that if the Lévy measure ν(dx), associated to the compensated Poisson random measure, satisfies ∫0<|x|≤1|x|ν(dx) < ∞, or ∫0<|x|≤1|x|2ν(dx) < ∞ and (in the second case) the Banach space is of type 2, then the pure jump martingale part in the decomposition is a stochastic integral of the function f(x) = x, in a stronger sense than in the decomposition given by Ito [Ito, K. On stochastic processes I (Infinitely divisible laws of probability). J. Math. 1942, 18, 261–301] resp. Dettweiler [Dettweiler, E. Banach space valued processes with independent increments and stochastic integrals. In Probability in Banach spaces IV, Proc., Oberwolfach 1982, Lectures Notes Maths., Springer: Berlin, 1982; 54–83], for the real resp. Banach valued case.