Mathematics for Economists

Optimal stopping

With backward iteration one should formulate the variables

Xt$max(Ht,E(Xt+ 1 j Ft)),

whereE(Xt+ 1 j Ft)is the conditional expectation ofXt+ 1 given

information available at timet.

It is not di¢ cult to show that

Xt=sup

τt

E(H(τ)j Ft)

That isXtis the best what one can get starting at timet.Xtis the same

as the value function in DP.

DeÖnition

Xtis called the Snell envelope.