Contingent claim pricing through a continuous time variational bargaining scheme

We consider a variational problem modelling the evolution with time of two probability measures representing the subjective beliefs of a couple of agents engaged in a continuous-time bargaining pricing scheme with the goal of finding a unique price for a contingent claim in a continuous-time financial market. This optimization problem is coupled with two finite dimensional portfolio optimization problems, one for each agent involved in the bargaining scheme. Under mild conditions, we prove that the optimization problem under consideration here admits a unique solution, yielding a unique price for the contingent claim.