Προβλήματα martingales και αλλαγές μέτρων
Martingale problems and changes of measure
Let (Ω;F; F; P) be a filtered probability space. Which are all the probability measures on F under which all the members of a given family X of processes are local martingales? Such a problem is called a martingale-problem. First ′′general′′ martingale problems are introduced, and then the martingale-problems related with point processes, random measures and semi-martingales are investigated. Next, the problem of what happens to a semi-martingale or a random measure, when one replaces the original measure P by another probability measure Q on F which is absolutely continuous with respect to P; is studied. Part of the problem consists in various versions of Girsanov’s theorem. Another part examined, consists in computing density processes. Solutions to the above problems have applications to actuarial science and financial mathematics.