The Kolmogorov–Čentsov theorem and Brownian motion in vector lattices |
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Authors: | JJ Grobler |
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Institution: | School for Computer, Statistical and Mathematical Sciences, North-West University, South Africa |
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Abstract: | The well known Kolmogorov–?entsov theorem is proved in a Dedekind complete vector lattice (Riesz space) with weak order unit on which a strictly positive conditional expectation is defined. It gives conditions that guarantee the Hölder-continuity of a stochastic process in the space. We discuss the notion of independence of projections and elements in the vector lattice and use this together with the Kolmogorov–?entsov theorem to give an abstract definition of Brownian motion in a vector lattice. This definition captures the fact that the increments in a Brownian motion are normally distributed and that the paths are continuous. |
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Keywords: | Vector lattice Stochastic process Hö lder-continuity Kolmogorov&ndash ?entsov theorem Brownian motion |
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