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.

Key Words:

• Stochastic integrals
• Random martingales measures
• Lévy–Ito decomposition
• Càdlàg processes on Banach spaces
• Independent increments
• Lévy measures
• Type 2 spaces

AMS Classification (2000):

• 60G51
• 60H05
• 47G30
• 46B09

Acknowledgment

We are very grateful to E. Dettweiler for kindly answering some questions we had concerning [13], V. Mandrekar for several very useful discussions concerning this work, L. Tubaro for clarifying comments during a workshop at Trento University (2002), and G. Ziglio for important comments to this work. We also thank J. Rosinski, and W. A. Woyczynski for pointing out to us [13] (cfr. Remark 4.8) and [33] [41] [47] [48] [59], and J. Rosinski for stimulating discussions. We are very grateful to O. Barndorff–Nielsen, T. Mikosch, E. Nicolato, G. Peskir, K. I. Sato and the Members of MaPhySto for inviting us to a most stimulating conference at Aarhus University (2002).

Notes

^#Dedicates this work to the memory of her father Ulrich Rüdiger.