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Unformatted text preview: x]. Doing the integrals and a little algebra gives
U ( x + (1 y )) U (y ) (U (x) U (y )) which after a little more algebra is (6.18).
Some concrete examples of utility functions are:
Up (x) = xp /p with 0 < p < 1 for x 0 and U (x) = U0 (x) = ln x for x > 0 and U (x) = 1 for x 0. Up (x) = xp /p with p < 0 for x 0 and U (x) = 1 for x < 0. 1 for x 0. 0
00
Dividing by p is useful here because in all three cases Up (x) = xp 1 and Up (x) =
p2
(p 1)x 0 for x 0. The second half of the deﬁnition is forced on us. In
0
the second and third cases Up (x) ! 1 as x # 0, while in the ﬁrst Up (x) ! +1
as x # 0. An optimal investment problem. Given a utility function U and an
initial wealth, ﬁnd an admissible trading strategy n to maximize EU (WN ),
where the wealth Wn satisﬁes the recursion (6.14)
Wn+1 = n Sn+1 + (1 + r)(Wn n Sn ) Example 6.6. Consider now the concrete example of maximizing U (x) = ln x
for a twoperiod binomial model with S0 = 4, u = 2, d = 1/2, and r = 1/4
where the probability of up and down are 2/3 and 1/3 respectively. Note that
we are optimizing under the real...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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