An optimal stopping problem with linear reward
Pierre van Moerbeke · Acta Mathematica · 1974
Background 2.1.The potential theory for the space-time brownian motion This potential theory was developed by J. L. Doob [8].The reader will find a short description of this theory in Ito-McKean [15].A real-valued function ] defined on R ~ is called excessive for the space-time brownian motion in an open domain D of R ~, if (i) ] is bounded below;(if) E/(x +x~, t +~)</(x, t) for every stopping time ~ not exceeding the first exit time VD from D.(iii) E/(x+xrn, t+T,)f ](x,t) for every sequence of stopping times ~,<~D such that P(~ ~ 0)= 1.This definition is equivalent to another in which (ii) and (iii) are replaced byNote that (iii') is superfluous if I is continuous.In such a case, exeessivity is a local property, indeed, a continuous function which is bounded below is excessive as soon as E/(x+z~ , t +~n) <-</(x, t)