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Unformatted text preview: itive, which implies that the log-optimal
investment problem is feasible. Show the following property: if there exists a v ∈ Rn with
1T v = 0, RT v RT v = 0 0, (40) then the log-optimal investment problem is unbounded (assuming that the probabilities pj are
(b) Derive a Lagrange dual of the log-optimal investment problem (or an equivalent problem of
your choice). Use the Lagrange dual to show that the condition in part a is also necessary for
unboundedness. In other words, the log-optimal investment problem is bounded if and only
if there does not exist a v satisfying (40).
(c) Consider the following small example. We have four scenarios and three investment options.
The return vectors for the four scenarios are 2 r1 = 1.3 ,
1 2 r2 = 0.5 ,
1 The probabilities of the three scenarios are
p1 = 1/3, p2 = 1/6,
108 0.5 r3 = 1.3 ,
p3 = 1/3, 0.5 r4 = 0.5 .
p4 = 1/6. The interpretation is as follows. We can invest in two stocks. The ﬁrst stock doubles in value
in each period with a probabi...
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- Fall '13
- The Aeneid