Characterization of stochastic control with optimal stopping in a Sobolev space

This work develops a new framework for a class of stochastic control problems with optimal stopping. One of our main motivations stems from dealing with the option pricing of American type. The value function is characterized as the unique solution of a partial differential equation in a Sobolev space. Together with certain regularities and estimates of the value function, the existence of the optimal strategy is established. The key ingredient is the use of the Itô formula for functions in a Sobolev space. Our approach provides a new alternative method for dealing with a class of stochastic control problems.