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Stochastic analysis without probability: study of some basic tools. (English) Zbl 1233.47036
In this article, authors propose a semi-group interpretation of essential notions and problems of stochastic analysis. By working on not necessarily Markovian semi-groups that arise from a big order generator, they define a generalized form of the Ito’s formula with respect to this generator. They also define the Wiener distribution with respect to a convolution semi-group and they prove that this generalized distribution is the solution to the martingale problem associated to the generator mentioned before.

MSC:
47D07 Markov semigroups and applications to diffusion processes
47D08 Schrödinger and Feynman-Kac semigroups
60G07 General theory of stochastic processes
60G20 Generalized stochastic processes
60G48 Generalizations of martingales
60H40 White noise theory
60J60 Diffusion processes
60J65 Brownian motion
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