Mathematics for Economists

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.