Degenerate Evolution Equations in Weighted Continuous Function Spaces, Markov Processes and the Black-Scholes Equation-Part II

In this second part of the paper, through applying semigroup theory procedures, we study initial boundary problems associated with degenerate second-order differential operators of the form Lu(x) ? α(x) u″(x)+β(x)u′(x)+γ(x) u(x) in the framework of weighted continuous function spaces on an arbitrary real interval, when particular boundary conditions are imposed. By using the general results stated in the first part, we show that such operators, frequently occurring in Mathematical Finance, generate positive strongly continuous semigroups, which are, in turn, the transition semigroups associated with suitable Markov processes. Finally, an application to the Black-Scholes equation is discussed, as well.