Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also?
That is, let $X_t^u$ is the solution to a controlled SDE $$ dX_t = \mu(t,u_t,X_t^u)dt + \sigma(t,u_t,X_t^u)dW_t , $$ $Y_t$ is a continuous stochastic process of $W_t$ and $J_t$ is the $\mathfrak{F}_t$-predictable loss functional $$ J_t(u_t)\triangleq \mathbb{E}\left[ \int_0^t \left( u_sX_s^u-Y_s \right)^2 ds \mid \mathfrak{F}_t^{X} \right]. $$ How can we solve for the optimal control $u_t^{\star}$ since $J_t$ is only measurable wrt $X$'s filtration and not $Y$'s?
