Applied Maths Seminar: Rutkowski -- Random Times and Martingales

The Applied Maths seminar this week is, as usual in the Access Grid Room at 2pm on
Wednesday.  The speaker is Marek Rutkowski on Random Times and Martingales.  The
abstract appears below.  

Please meet at Carslaw level 6 lifts at 12.30pm to go to lunch with the speaker.  

Title: Random Times and Martingales 

Abstract: 

Random times were object of numerous studies during the last fifty years but, somewhat
surprisingly, their fine properties are still not fully understood.  As opposed to the
so-called stopping times, general random times are used to model additional randomness,
which is not covered by observations of a reference stochastic process, for instance,
for insider trading or credit risk.  Hence a detailed study of various forms of the
dependence between a random time and a reference stochastic process is a crucial step in
most applications of random times.  The Azema supermartingale and the jump martingale
associated with a random time appear to be important tools in this study.  

We start by presenting a brief overview of concepts related to constructions and
classification of random times.  Subsequently, we will focus on the progressive
enlargement of a reference filtration with a random time and we will study the case
where the avoidance hypothesis fails hold.  We first present some basic examples where
the classic (H’) hypothesis is trivially satisfied, for instance, when the reference
filtration is generated by a Poisson process.  Our ultimate goal is to check whether the
predictable representation property for martingales, which is assumed to hold in the
reference filtration, is also satisfied in the progressively enlarged filtration.