A primer on stochastic control and portfolio optimization

We will give a short introduction to stochastic control in continuous time with some
applications to optimal investment problems. In particular, after giving some examples of
control problem which are relevant in finance and insurance, we will turn to the dynamic
programming principle (DPP), which is the main tool to obtain the Hamilton-Jacobi-Bellman
(HJB) partial differential equation describing the local behaviour of the value function. Under
some regularity conditions, solving this HJB pde gives a method to find (at least theoretically)
the optimal solution. We will apply this approach to solve some problems of optimal
investment, e.g. the classic Merton problem of optimal investment and consumption.