By Dimitri P. Bertsekas

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X, to be invested. ,ai2 m/A. i= 1 This problem may be solved for A = 1 and various values of m. ” Given a pair { 1 IT 1, j } on this locus it is easy to see that the corresponding standard deviation and mean associated with a total investment of A monetary units will be A IIT I and A j , respectively. In this way a mean standard deviation locus constructed for a single level of total investment may be used for any level of investment. Thus the approach allows a once and for all computation, valid for every level of investment and hence for every investor.

N - 1, 9 and any other existing constraints on the feedback controller. We note that the description of the problem given above is intended to be informal. A precise definition together with examples will be provided in the next chapter. Dynamic programming is directly applicable to problems of this form. Furthermore, as will be shown later, various other dynamic optimization problems involving, for example, correlated disturbances, imperfect state information, and nonadditive cost function, can be directly reformulated into the problem described above (perhaps by introducing a more complex 1 32 INTRODUCTION structure).

Consider a roulette wheel that is partitioned into k disjoint events A , , . . , Ak such that the probability of occurrence of A i is pi,i = 1 , . . , k, and pi = 1. Suppose a person is given an amount of x dollars that he can allocate among the k events A , , . . ,A , and that he receives as his reward the amount x i that he has allocated to the event A i that actually occurs. The constraints are x i 2 0, x i = x. Suppose that his utility function over If= If=l 36 1 INTRODUCTION positive monetary rewards r is U(r) = In r.

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