On some non-linear boundary value problems related to a Black--Scholes model with transaction costs

We deal with some generalizations on a Black--Scholes model arising in financial mathematics. As novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large class of problems involving Black--Scholes equation. Our main results involve the existence of extremal solutions in presence of lower and upper solutions. Some examples of application are provided too.