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

Abstract

We consider the solvability of the ordinary differential equation (ODE)

inside an interval , where σ, 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 Itô 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.

Keywords:

• Second-order linear ordinary differential equations
• Measure-valued inhomogeneity
• Additive functionals
• Local time
• Optimal stopping
• Singular control

2000 Mathematics Subject Classification::

• 34A30
• 60J55
• 49J15
• 34H05
• 62L15

Acknowledgements

We are grateful to several participants in the Optimal Stopping with Applications Symposium that was held in Manchester on 22–27 January 2006, where part of the results in this paper were presented by the second author, for numerous helpful discussions. We are indebted to Andrew Jack and Damien Lamberton whose critical and detailed comments lead to a significant enhancement of the paper. We also thank an anonymous referee for several useful comments.

Notes

^§ [email protected]

^†Research supported by EPSRC grant no. GR/S22998/01.

Additional information

Notes on contributors

Timothy C. Johnson

§ § [email protected]