A domain-theoretic approach to Brownian motion and general continuous stochastic processes
File(s)
Accepted version
Author(s)
Bilokon, P
Edalat, A
Type
Journal Article
Abstract
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 stochas-
tic processes and characterise their computability. For a given continuous stochastic
process, we show how its domain-theoretic, i.e., finitary, approximations can be con-
structed, 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 de-
fined Wiener measure and show that the Wiener measure is computable within the
domain-theoretic framework.
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 stochas-
tic processes and characterise their computability. For a given continuous stochastic
process, we show how its domain-theoretic, i.e., finitary, approximations can be con-
structed, 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 de-
fined Wiener measure and show that the Wiener measure is computable within the
domain-theoretic framework.
Date Issued
2017-08-09
Date Acceptance
2017-05-12
Citation
Theoretical Computer Science, 2017, 691, pp.10-26
ISSN
0304-3975
Publisher
Elsevier
Start Page
10
End Page
26
Journal / Book Title
Theoretical Computer Science
Volume
691
Copyright Statement
© 2017, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Subjects
08 Information And Computing Sciences
01 Mathematical Sciences
Computation Theory & Mathematics
Publication Status
Published