Hedging under illiquidity costs as a second order target problem: a dual formulation

Starting from the problem of perfect hedging under market illiquidity, as introduced by Cetin, Jarrow and Protter, we introduce a class of second order target problems. A dual formulation in the general non-Markov case is obtained by formulating the problem under a convenient reference measure. In contrast with previous works, the controls lie in the classical H2 spaces associated to the reference measure. A dual formulation of the problem in terms of a standard stochastic control problem is derived, and involves control of the diffusion component.