Before revision:  By proving a martingale representation theorem based on incomplete and phasing out filtration, we characterize the value function in an equivalent form. This allows us to connect the control problem to a quadratic BSDE with jumps, in which the driver term has non-standard structures and carries the conditional filter as an infinite-dimensional parameter.
After revision:  By proving a martingale representation theorem based on incomplete and phasing out filtration, we connect the control problem to a quadratic BSDE with jumps, in which the driver term has non-standard structures and carries the conditional filter as an infinite-dimensional parameter.