Mathematics for Economists

Optimal stopping

Obviously

VT(ξ)=ξmax



ξ,E(VT^1 (ξ))

1 +r



=VT 1 (ξ),

With induction

Vt+ 1 (ξ)=max



ξ,

E(Vt+ 2 (ξ))

1 +r



max



ξ,

E(Vt+ 1 (ξ))

1 +r



=Vt(ξ),

hence

αt+ 1 =E(Vt+^2 (ξ))

1 +r

E(Vt+^1 (ξ))

1 +r

=αt.

DeÖnition

(αn)is the optimal exercise boundary.