The solution to a second order linear ordinary differential equation with a non-homogeneous term that is a measure

We consider the solvability of the ordinary differential equation (ODE)[image omitted]inside an interval [image omitted], where , b, r are given functions and h is a locally finite measure. This ODE is associated with the Hamilton-Jacobi-Bellman (HJB) equations arising in the study of a wide range of stochastic optimisation problems. These problems are motivated by numerous applications and include optimal stopping, singular stochastic control and impulse stochastic control models in which the state process is given by a one-dimensional It diffusion. Under general conditions, we derive both analytic and probabilistic expressions for the solution to equation (1) that is required by the analysis of the relevant stochastic control models. We also establish a number of properties that are important for applications.