Mathematics for Economists

Optimal stopping

Example

Buying a stock with independent o§ers.

The Snell envelope is

XT = HT$ξT

Xn = min(ξn,E(Xn+ 1 j Fn))

As the future is independent from the presentE(Xn+ 1 j Fn)=E(Xn+ 1 ).

Xn=min(ξn,E(Xn+ 1 )),

hence ifαn$E(Xn+ 1 )then the optimal strategy is

τ=minfn 0 jξnαng^T.

At timeT one must buy the stock.. At timeT1 one must buy it if

ξT 1 is smaller than the expected value. etc.