An optimal stopping problem with finite horizon for sums of I.I.D. random variables

Abstract

AbstractThe problem of selling a commodity optimally at one of n successive time instants leads to the optimal stopping problem for the finite sequence ((n−j)lSj)1⩽j⩽n, where Sj=U1 + … + Uj, U1, U2,… are i.i.d., E(U1) = 0 and E(U21) = 1. The optimal stopping time πn is seen to be of the form τn = inf{j|j = n or j < n, Sj⩾clj,n}, where c1j,1>…>cln−1,n = 0 satisfyn−12 cj,nl → αl(1 − t)11, if n → ∞, j/n →t ṫ[0,1]. αl > 0 is the solution of the equation d2l+2dx2l+2(Ф/φ)(α) = (α + α−1)d2l+2dx2l+2(Ф/φ)(α). For the value vln we have n−32vnl → vl. vl is explicitly computed. In the normal case we also obtain results on the speed of convergence of n−12cj,nl and n−32vnl