Piecewise constant martingales and lazy clocks

Abstract

This paper discusses the possibility to find and construct 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 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 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