# Piecewise Constant Martingales and Lazy Clocks

## TL;DR

This paper introduces a novel method for constructing piecewise constant martingales using lazy clocks, which are time-change processes that synchronize with real time at random moments, enabling analytical tractability.

## Contribution

The paper proposes a new construction of lazy martingales via lazy clocks, allowing for piecewise constant martingales without assumptions on the latent process.

## Key findings

- Lazy clocks can synchronize with real time at random moments.
- The method produces analytically tractable piecewise constant martingales.
- Lazy martingales are adaptable to their filtration, unlike standard time-changed processes.

## Abstract

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 $\mathbb{R}$. After a brief review of standard possible techniques, we propose a construction based on the sampling of latent martingales $\tilde{Z}$ with \textit{lazy clocks} $\theta$. These $\theta$ 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 $Z_t=\tilde{Z}_{\theta_t}$ a martingale (called a \textit{lazy martingale}) without any assumptions on $\tilde{Z}$, and in most cases, the lazy clock $\theta$ is adapted to the filtration of the lazy martingale $Z$. This would not be the case if the stochastic clock $\theta$ 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) $\mathbb{R}$.