×

Filtered Azéma martingales. (English) Zbl 1317.60049

Summary: We study the optional projection of a standard Brownian motion on the natural filtration of certain kinds of observation processes. The observation process, \(Y\), 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 \(Y\) hits 0. As such, the associated optional projections are related to Azéma’s martingales which are obtained by projecting the Brownian motion onto the filtration generated by observing its signs.

MSC:

60G44 Martingales with continuous parameter
60G35 Signal detection and filtering (aspects of stochastic processes)
60J65 Brownian motion
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J55 Local time and additive functionals