Optimal stopping
The expected gain with this strategy
( 1 +r)E
max
1
4
,ξ 2
= 2
Z 1 / 4
0
1
4
dx+
Z 1
1 / 4
xdx
=
=^1
8
+ 2
x^2
2
1
1 / 4
=^1
8
+ 1 ^1
16
=^17
16
=
=^1
4
^1
2
+^3
4
2 ^5
8
=
=^1
4
E(ξ 3 )+^3
4
( 1 +r)^5
8
.
Where 5/8 is the expected value of the uniform distribution on[ 1 / 4 , 1 ].