Optimal stopping
Substituting back and recalling that
h(x)$
E(max(x,ξT))
1 +r =
E(max(η,ξT)jη=x)
1 +r
On the set
Sn=Sn(T)=fmaxknξkag=fmaxknξkh(maxknξk)g
Vn=max
maxknξk,E(maxkn+^1 ξkj Fn)
1 +r
=
=max
maxknξk,
E(max(maxknξk,ξn+ 1 )j Fn)
1 +r
$
$max
maxknξk,h
maxknξk
=maxknξk,
SoSn(T)Sn(T+^1 )hence the setfHnXngis independent of the number
of periods left.