Discrete stopping times in the lattice of continuous functions

Abstract

A functional calculus for an order complete vector lattice $\mathcal{E}$ was developed by Grobler in 2014 using the Daniell integral. We show that if one represents the universal completion of $\mathcal{E}$ as $C^{\infty }(K)$, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in $C^{\infty }(K)$. This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in $C^{\infty }(K)$. We obtain a representation that is analogous to what is expected in probability theory.