Suppose there is a stochastic optimal control problem where the uncertainty is because of the random arrival of entities into our system which is modelled as a Poisson process with rate $\lambda(p)$. The BHJ optimality equation is as follows:
where $(x,q,t)$ are states of our system. Since the planning horizon is finite, $t\in [0,T]$, I should consider time as a state. $V(x,q,t)$ is the cost-to-go when the system is in state $(x,q,t)$. Since the system might be noisy, we may access some realization of $\lambda(p)$, and the exact value is not known. I am wondering if there is any data-driven approach to solve this BHJ equation?
I should also mention that the state-space can be extremely large and approximation might be used. I would be thankful if you can recommend some reference to handle this problem.