Guaranteed Deterministic Approach to Superhedging: Lipschitz Properties of Solutions of the Bellman–Isaacs Equations
For the discrete-time superreplication problem, a guaranteed deterministic formulation is proposed: the problem is to ensure the cheapest coverage of the contingent claim on an American option under all admissible scenarios. These scenarios are set by a priori defined compacts depending on the price history; the price increment at each moment of time must lie in the corresponding compact. The market is considered without trading constraints and transaction costs. The problem statement is game-theoretic in nature and leads directly to the Bellman–Isaacs equations of a special form under the assumption of no trading constraints. In the present study, we estimate the modulus of continuity of uniformly continuous solutions, including the Lipschitz case.