Partially informed investors: hedging in incomplete market with default

In a defaultable market, an investor trades having only a partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modeled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, this note examines stochastic control problems dealt with Backward Stochastic Differential Equations, (BSDEs), and filtering techniques. The goal of this note is to construct a sequence of functions converging to the value function, each of these is the unique solution of suitable BSDEs.