# How to solve Bellman-Hamiltonian-Jacobi in stochastic environment through machine learning?

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.

• I don't really understand this kind of stuff but it looks a bit like an optimization problem, right? If so genetic learning might be applicable. Sep 22 '21 at 16:17