Mathematics for Economists

Optimal stopping

As we have already seen ifT=n+ 1 ,thenSn(T)=fmaxknξkag.

Now letT=n+2 that is two periods are still ahead. We must show that

Sn=Sn(T)Sn(T+^1 ).

Vn$max



maxknξk,E(Vn 1 ++^1 j Fr n)



=

=max

0

@max

kn

ξk,

E



max



maxkn+ 1 ξk,E(Vn+ 12 +jFrn+^1 )



j Fn



1 +r

1

A.

As

max

kn+ 1

ξkmax

kn

ξka

using the induction hypothesis that if one periods left then the

fxjxagis the stopping region

max



kmaxn+ 1 ξk,E(Vn+^2 j Fn+^1 )

1 +r



=kmaxn+ 1 ξk.