Solution of the Optimal Stopping Problem for One-Dimensional Diffusion Based on a Modification of the Payoff Function

A problem of optimal stopping for one-dimensional time-homogeneous regular diffusion with the infinite horizon is considered. The diffusion takes values in a finite or infinite interval ]a,b[. The points a and b may be either natural or absorbing or reflecting. The diffusion may have a partial reflection at a finite number of points. A discounting and a cost of observation are allowed. Both can depend on the state of the diffusion. The payoff function g(z) is bounded on any interval [c, d], where a < c < d < b, and twice differentiable with the exception of a finite (may be empty) set of points, where the functions g(z) and \({g}^{{^\prime}}(z)\) may have a discontinuities of the first kind. Let L be an infinitesimal generator of diffusion which includes the terms corresponding to the discounting and the cost of observation. We assume that the set \(\{z : Lg(z) > 0\}\) consists of a finite number of intervals. For such problem we propose a procedure of constructing the value function in a finite number of steps. The procedure is based on a fact that on intervals where Lg(z) > and in neighborhoods of points of partial reflections, points of discontinuities, and points a or b in case of reflection, one can modify the payoff function preserving the value function. Many examples are considered.