Partially Controllable Random Walk

摘要

This paper deals with an almost uncontrollable process which will be termed a partially controlled random walk. A particle performs a random walk on an interval (o, a). A utility U sub o (x) is associated with each position x on (o, a). The amplitude of the random walk is controllable at each stage. The process is termed almost uncontrollable because the random walk is assumed to have zero mean. Gambling terminology is used because of its extensive vocabulary, but the utility functions U sub o (x) can be nonmonotonic and discontinuous--more general functions than their interpretation as gambling utilities would suggest. This problem may be viewed as a sequence of fair gambles on a nonlinear utility function. It is shown that with certain betting sequences a certain higher utility U(x) can be associated with each point x, where U(x) is now the expected utility at the end of the betting sequence beginning with capital x. An upper bound on U(x) (over all possible infinite betting sequences) will be derived. This maximum expected utility U(x) has a very nice relation to U sub o (x), the actual utility.