Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints

We study the problem of minimal initial capital needed in order to hedge a European contingent claim without risk. The financial market presents incompleteness arising from two sources: stochastic volatility and portfolio constraints described by a closed convex set. In contrast with previous literature which uses the dual formulation of the problem, we use an original dynamic programming principle stated directly on the initial problem, as in Soner and Touzi (1998. SIAM J. Control Optim.; 1999. Preprint). We then recover all previous known results under weaker assumptions and without appealing to the dual formulation. We also prove a new characterization result of the value of super-replication as the unique continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a suitable terminal condition.