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Lévy processes and stochastic integrals in Banach spaces

100%

Applebaum D.

Probability and Mathematical Statistics

2007

tom Vol. 27, Fasc. 1

75--88

We review infinite divisibility and Lévy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Lévy-Itô decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct generalised Ornstein-Uhlenbeck processes

A Levy-Ciesielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges

100%

Applebaum D.

Journal of Applied Analysis

2007

tom Vol. 13, nr 2

275-290

We introduce "probabilistic" and "stochastic Hilbertian structures". These seem to be a suitable context for developing a theory of "quantum Gaussian processes". The Schauder system is utilised to give a Levy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.