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 martingales Z with 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 (calendar) clock.This specific choice makes the resulting time-changed process Zt = Zθt a martingale (called a lazy martingale) without any assumption on Z, and in most cases, the lazy clock θ is adapted to the filtration of the lazy martingale Z,so that sample paths of Z on [0, T] only requires sample paths of (θ,Z) up to T.This would not be the case if the stochastic clock θ could 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.
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