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.
CITATION STYLE
Bilokon, P., & Edalat, A. (2017). A domain-theoretic approach to Brownian motion and general continuous stochastic processes. Theoretical Computer Science, 691, 10–26. https://doi.org/10.1016/j.tcs.2017.07.016
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