In typical stochastic control problems, the control variables in HJB equation are usually continuous or discrete values in action space. However, what if the action space is constructed by unknown continuous functions?
For example, in market making problem, the Avellaneda-Stoikov model calculates the optimal quoted bid and ask prices, while fixing their volumes. If we assume the bid-ask prices and the quoted volumes are continuous, what’s the optimal quoted volumes (V) along the quoted prices (x)? In such a case, we need to solve the function V(x) on both bid and ask sides. Is there any HJB-like method to solve it?