Slightly inaccurate observations in continuous time
Francois Coquet, Ph.D.
Department of Statistics , Ecole Nationale de la Statistique et de l'Analyse de l'Information (ENSAI), France
March 31, 2008
Stochastic processes in continuous time are still an increasingly
popular tool to deal with problems arising from physics, finance or
biology. However, in practice, it is impossible to effectively observe
the whole trajectory of such a process: at best you can observe a
sample at a discrete set of times, moreover any observation will be
subject to round-off errors. So a question arises about the accuracy of
observations regarding the model they account for.
It is generally believed that if the sampling scheme is sufficiently narrow, and every observation sufficiently precise, the inaccuracy of information involved in this approximation becomes negligible. We will show that it may be much more tricky, and that very close (in any sense) processes may carry very different information.
This will be illustrated through an optimal stopping problem arising from finance, and we will give some criteria insuring that approximations close to the model do carry close information.