Piecewise constant martingales and lazy clocks

Christophe Profeta; Frédéric Vrins

doi:10.1186/s41546-019-0036-4

Article Contents

2019, Volume 4: 2. Doi: 10.1186/s41546-019-0036-4

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Piecewise constant martingales and lazy clocks

Christophe Profeta¹ and
Frédéric Vrins^2, ,

1. Université d'Évry, France;
2. Louvain Finance Center(LFIN) and Center for Operations Research and Econometrics(CORE), Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium

Frédéric Vrins, frederic.vrins@uclouvain.be

Received: February 28, 2018

Revised: January 16, 2019

Published: July 2019

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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 $\mathbb{R}$ 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 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 Z_t = $\tilde Z$_{θ_t} a martingale (called a lazy martingale) without any assumption on $\tilde Z$, 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, $\tilde Z$) 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) $\mathbb{R}$.

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