STOCHASTIC

(b) Interpret this assumption. Is it strong or weak?
(c) Show that Breiman's assumption that the random returns are finite and bounded
away from zero is sufficient but not necessary for the satisfaction of the boundedness
assumption.
(d) Use the Chebychev inequality to prove that for independent returns, a max E log VN
criterion has a modified version of Property 1 (where convergence in probability replaces
almost sure convergence) stated in Thorp's paper. Assume that all feasible strategies
are bounded.
(e) Use the Chebychev inequality to prove that for dependent returns, a max E log VN
criterion has the modified Property 1. Assume that all strategies under consideration
are significantly different and bounded, and that the return distributions are independent
of the investor's strategy.
(f) What can be said about Property 2 in Thorp's paper ?
•24. Refer to the papers by Thorp and Breiman.
(a) Attempt to prove that Thorp's Property 2 does hold for Breiman's model.
(b) Attempt to prove that for a sequence of investment choices Thorp's Property 1
holds if and only if Property 2 holds.
(c) Assume that the investor's initial wealth is the positive integer M, that only integervalued
returns are possible, and that the X, must be integers. Suppose In* is a maximum
expected log strategy. Attempt to show that Theorems 1-3 apply to 1N*.
(d) Does Property 2 hold for this model ?
25. (The Kelly criterion and expected utility) Suppose an investor's utility function is
WT" over period T's wealth WT, where a < 1 and a ^ 0. Let the gross rate of return in each
period t be r,, where r, = £ rt Xit, the X„ are the relative investment allocations in period t,
and the rt are the gross rates of return for the individual investments assumed to be independently
and identically distributed in time. Suppose Wa > 0 is the investor's initial wealth;
then his wealth at time 7" is
WT= Waf[rx.
t-i
(a) Show that one may maximize WT for fixed r, by choosing the X„ so that in each
period they maximize the expected logarithm of one period return, namely,
max log ][>,*„, s.t. ~£Xlt = 1, X„ 5; 0.
i i
(b) Show that the optimal strategy, say A", is stationary in time, i.e., X' — (Xlt, ...,Xm)
for all t.
(c) Show that with probability 1 the return under the X' strategy is at least as high as
under the strategy X p , where X p is the solution to
max£Q>,AY)