Piecewise constant martingales and lazy clocks

Conditional expectations(like,e.g.,discounted prices in financial applications)are martingales under an appropriate filtration and probability measure.When the information flow arrives in a punctual way,a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates.Providing a way to find and construct piecewise constant martingales evolving in a connected subset of R is the purpose of this paper.After a brief review of possible standard techniques,we propose a construction scheme based on the sampling of latent martingalesZ with lazy clocksθ.Theseθare time-change processes staying in arrears of the true time but that can synchronize at random times to the real(calendar)clock.This specific choice makes the resulting time-changed process Zt=Zθt a martingale(called a lazy martingale)without any assumption onZ,and in most cases,the lazy clockθis adapted to the filtration of the lazy martingale Z,so that sample paths of Z on[0,T]only requires sample paths ofθ,Zup to T.This would not be the case if the stochastic clockθcould be ahead of the real clock,as is typically the case using standard time-change processes.The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on(interval of)R.