Piecewise constant martingales and lazy clocks

Abstract

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