Optimal Stopping of Integral Functionals and a “No-Loss” Free Boundary Formulation

This paper is concerned with a modification of the classical formulation of the free boundary problem for the optimal stopping of integral functionals of one-dimensional diffusions with, possibly, irregular coefficients. This modification was introduced in [L. Rüschendorf and M. A. Urusov, Ann. Appl. Probab., 18 (2008), pp. 847–878]. As a main result of that paper a verification theorem was established. Solutions of the modified free boundary problem imply solutions of the optimal stopping problem. The main contribution of this paper is to establish the converse direction. Solutions of the optimal stopping problem necessarily also solve the modified free boundary problem. Thus the modified free boundary problem is also necessary and does not “lose” solutions. In particular, we prove smooth fit in our situation. In the final part of this paper we discuss related questions for the viscosity approach and describe an advantage of the modified free boundary formulation.