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

The 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[less-than-or-equals, slant]j[less-than-or-equals, slant]n, where Sj=U1 + ... + Uj, U1, U2,... are i.i.d., E(U1) = 0 and E(U21) = 1. The optimal stopping time [pi]n is seen to be of the form [tau]n = inf{jj = n or j < n, Sj[greater-or-equal, slanted]clj,n}, where c1j,1>...>cln-1,n = 0 , if is the solution of the equation . For the value vln we have . vl is explicitly computed. In the normal case we also obtain results on the speed of convergence of and .