Stochastic exit-time control on the half-line over a finite horizon

Abstract

We consider a finite-time stochastic drift control problem with the assumption that the control is bounded and the system is controlled until the state process leaves the half-line. Assuming general conditions, it is proved that the resulting parabolic Hamilton-Jacobi-Bellman equation has a classical solution. In fact, we consider an even more general family of semilinear equations, which might be helpful in solving other control or game problems. Not only is the existence result proved, but also a recursive procedure for finding a solution resulting from a fixed-point argument is provided.

This paper supersedes arXiv:1703.07339, [v2] - major revision, 26 pages, [v3] - Version accepted for publication in Annales Polonici Mathematici (pagination may differ due to formatting); Sections 6 and 7 of [v2] are left for future work