Background :
I have studied mathematics and have a good knowledge of stochastic processes and a fairly good knowledge of stochastic calculus (for this I am currently reading Brownian Motions, Martingales and Stochastic Calculus, which I find a well-written introduction to the subject.
Type Of Problem I would like to solve :
I'm interested in theoretical inventory theory. The general idea is the following: we have a shop which sells a certain product, demand arrives according to some (continuous) stochastic process. As the demand depletes the inventory of the shop, the shop has to order new supplies, these supplies then arrive after some time. As long as the shop has supplies a holding cost of $h$ per unit is incurred and if the shop runs out of stock, it gets a negative stock with a backlog cost $b$ per backlogged unit.
We would now like to descibe which ordering policy would be optimal for this shop (for this example this problem has of course already been solved, but I would like to solve it for similar problems).
See for example the Inventory Management with Stochastic Lead Times.
Needed Reference :
I have been looking around on how to tackle this problem and it seems like this is a ''stochastic control'' problem. For this I found the book stochastic controls: Hamiltonian systems and HJB equations, which at first sight looks pretty good. But as I am just getting into the problem, I would like to hear from you other (better?) references.
There are a lot of books on inventory theory but they all seem very "low level" to me, not dealing with anything like stochastic differential equations etc.