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 Z̃ 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 Z. = Z̃(θ.) a martingale (called a lazy martingale) without any assumptions on Z̃, 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.
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https://hal.archives-ouvertes.fr/hal-01537241
Contributor : Christophe Profeta <>
Submitted on : Monday, June 26, 2017 - 4:43:08 PM
Last modification on : Friday, July 20, 2018 - 11:13:01 AM
Long-term archiving on : Tuesday, December 12, 2017 - 10:49:24 AM

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  • HAL Id : hal-01537241, version 1

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Christophe Profeta, Frédéric Vrins. Piecewise Constant Martingales and Lazy Clocks. 2017. ⟨hal-01537241⟩

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