A domain-theoretic approach to Brownian motion and general continuous stochastic processes

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Theoretical Computer Science, ISSN: 0304-3975, Vol: 691, Page: 10-26

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Paul Bilokon, Abbas Edalat
Elsevier BV
Mathematics, Computer Science
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We introduce a domain-theoretic framework for continuous-time, continuous-state stochastic processes. The laws of stochastic processes are embedded into the space of maximal elements of the normalised probabilistic power domain on the space of continuous interval-valued functions endowed with the relative Scott topology. We use the resulting ω -continuous bounded complete dcpo to obtain partially defined stochastic processes and characterise their computability. For a given continuous stochastic process, we show how its domain-theoretic, i.e., finitary, approximations can be constructed, whose least upper bound is the law of the stochastic process. As a main result, we apply our methodology to Brownian motion. We construct a partially defined Wiener measure and show that the Wiener measure is computable within the domain-theoretic framework.

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