In this paper, we consider the problem of super-replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super-replication value is the smallest function which lies above the Black-Scholes price function and which is stable for the so-called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.
Numerical Analysis and Computation | Probability
Bensoussan A., Touzi N. and Menaldi J.-L. (2005). Penalty approximation and analytical characterization of the problem of super-replication under portfolio constraints, Asymptotic Analysis 41, 311-330.