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Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula
Author(s) -
Karlsson John,
Löbus JörgUwe
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500146
Subject(s) - mathematics , ornstein–uhlenbeck process , quadratic variation , dirichlet form , mathematical analysis , quadratic equation , diffusion process , classical wiener space , scalar (mathematics) , dirichlet distribution , integral representation theorem for classical wiener space , wiener process , stochastic process , brownian motion , geometry , integral equation , functional integration , knowledge management , statistics , innovation diffusion , computer science , boundary value problem
The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron–Martin space. It is shown that the distributions of certain finite dimensional Ornstein–Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein–Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.