TL;DR
This paper investigates the optional projection of Brownian motion onto filtrations generated by observation processes that reveal sign information, linking to Azéma's martingales, and explores their properties.
Contribution
It introduces a new class of filtered Azéma martingales derived from stochastic differential equations involving sign information of Brownian motion.
Findings
Characterization of the optional projection process
Connection between observation processes and Azéma's martingales
Insights into the structure of filtered martingales
Abstract
We study the optional projection of a standard Brownian motion on the natural filtration of certain kinds of observation processes. The observation process, , is defined as a solution of a stochastic differential equation such that it reveals some (possibly noisy) information about the signs of the Brownian motion when hits 0. As such, the associated optional projections are related to Az\'ema's martingales which are obtained by projecting the Brownian motion onto the filtration generated by observing its signs.
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