This paper discusses the possibility to find and construct \textit{piecewise
constant martingales}, that is, martingales with piecewise constant sample
paths evolving in a connected subset of R. After a brief review of
standard possible techniques, we propose a construction based on the sampling
of latent martingales Z~ with \textit{lazy clocks} θ. These
θ are time-change processes staying in arrears of the true time but that
can synchronize at random times to the real clock. This specific choice makes
the resulting time-changed process Zt=Z~θt a martingale
(called a \textit{lazy martingale}) without any assumptions on Z~, and
in most cases, the lazy clock θ is adapted to the filtration of the lazy
martingale Z. This would not be the case if the stochastic clock θ
could be ahead of the real clock, as typically the case using standard
time-change processes. The proposed approach yields an easy way to construct
analytically tractable lazy martingales evolving on (intervals of)
R.Comment: 17 pages, 8 figure