Abstracts

We consider a probability space equipped with two filtrations F and G, one contained in the other. The theory of enlargement of filtration is the research about the relationship between the F-semimartingales and the G-semimartingales. It provides particular techniques to do the stochastic calculus. This theory plays an essential role in credit risk modeling. It also constitutes the major instrument to understand the dependence of a market on the change of information and to inspect the non-arbitrage property. There are two well-known and distinct theories called initial enlargement of filtration and the progressive enlargement of filtration. Such a distinction is the natural consequence of the disparity between the techniques dealing with these two cases, and becomes the standard in the applications of market modeling. However, there exists important models which are covered by neither initial enlargement theory nor the progressive enlargement theory, such as the model of the enlargement by future infima of a positive diffusion. In this talk we present a general theory, called local solution method, to deal with the enlargement of filtration, which contains the initial enlargement and the progressive enlargement theories as corollaries. The local solution method is based on the following observation. The problem of the enlargement of filtration can be defined and be studied locally at every point much like the notion of the derivative defined and computed for a function. In the same time, the problem of the enlargement of filtration also is a global problem much like the integration of a function. The local solution method provides techniques to make use of these two aspects of the problem to yield corresponding solutions. Examples will be given to illustrate how the local solution method constitutes an effective and flexible method. Notably the initial enlargement and the progressive enlargement theories will be investigated anew with the local solution method as well as the model of the enlargement by future infima of a positive diffusion.