A guaranteed deterministic approach to superhedging: the proprieties of semicontinuity and continuity of the Bellman–Isaacs equations

A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure the coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The absence of transaction costs is assumed; the market is considered with trading constraints. The game-theoretical interpretation implies that the corresponding Bellman–Isaacs equations holds. In the present paper we propose several conditions for the solutions of these equations to be semicontinuous or continuous.