ABSTRACT: I shall discuss certain techniques in free boundaries related to American type contracts. Such techniques are of general nature and have the advantage of being applied to general framework in applications. Our focus shall be on the behavior of the solution function as well as the free boundary, close to initial state (American Option) or close to Dirichlet data (Convertible Bonds).

ABSTRACT: In this talk I shall discuss 2-phase Quadrature domains, and some aspects related to moving boundaries. I shall also mention possible generalizations (as toy models) of the concept into multi-phase case. Seen, as free boundaries, I will also mention aspects of regularity theory, uniqueness and existence. At this stage of the development of the theory, there are more questions, than answers. I shall also connect to Combinatorial Random Walks as discrete version of this problem. One such example is the famous Diaconis-Fulton Smash Sum.