Risk minimizing hedging under partial information via Backward Stochastic Differential Equations

A financial market in the case where there are restrictions on the available is considered. We provide the Galtchouk-Kunita-Watanabe representation for a contingent claim under restricted information and, as a consequence, we deduce existence and uniqueness for the solution of linear backward stochastic differential equations (BSDEs) driven by a general càdlàg martingale in a partial information setting. Next, this result is extended to non linear BSDEs with Lipschitz driver and we provide the Föllmer-Schweizer decomposition (in the restricted information framework) with respect to an underlying risky asset price process described by a semimartingale. We discuss an application to risk-minimization. First, in the case where the risky asset price process is directly modeled under a martingale measure. Second, in the more general semimartingale case by introducing the concept of locally risk-minimizing strategies and characterizing the optimal strategy via the Föllmer Schweizer decomposition under restricted information. Finally, an example in the martingale case shows how to compute the risk-minimizing hedging strategy in terms of the filter when the risky asset price is described by a jump-diffusion process. More precisely, we assume that the behavior of the risky asset price depends on an unobservable stochastic factor and that the investors can only observe the prices but not the stochastic factor which affects their dynamics.