Stopping on patterns in strings and applications to optimal search and selection problems

This talk presents an attempt to unify sequential search and selection problems in the light of optimal stopping problems on strings. The basic model is as follows: Strings are generated by sequences of letters or other codes produced by independent draws from a given alphabet or another source. The law of drawing different letters or expressions may be time-invariant (stationary) or else depend on time. For a fixed n and a given pattern H = H1H2, · · ·Hl of length l ≤ n our goal is to maximize the probability of stopping on the kth last appearance of H (if any) in such a string of length n, given that we must not return on a previous appearance of H. We shall easily see that a first approach to this problem can be achieved by Aldous’ (1989) Poisson clumping heuristic. However this elegant approach, being a limit approach, is shown to be too coarse for our purposes and asks for a more precise solution. Our problem is a generalization of the problem of stopping on sequences of independent indicator functions (Bruss(2000)). The solution of that problem, the so-called odds-algorithm, seems to play an essential role in this general problem as well. Indeed, we can prove this in the stationary case for so-called nonautocorrelated patterns. There are some essential differences to the case l = 1, which make the new problem, in a certain way, more attractive for possible applications. This is, in particular, the fact that the typical lower success probability bound 1/e for the case l = 1 can now be shown to have a sibling in form of a typical upper bound, which can, depending on the length and form of H, be as large as 0.619 · · ·, and this for quite realistic problems. The results are in part joint work with Guy Louchard (ULB, Brussels). We also look at a few elementary models for investment under this light. The basic question is: How should we invest capital into a sequence of investment opportunities, if we want to invest in the very last, respectively, in a pre-specified number of last investment opportunities in a given horizon. Viewing high-risk situations we assume that an investment on the very best opportunity yields a lucrative, possibly time-dependent, rate of return, that uninvested capital keeps its risk-free value, whereas “wrong” investments lose their value. Several models are presented, mainly for the so-called rank-based case, and optimal strategies and values are found. In particular, we are interested in tractable models for an unknown number of opportunities. very best opportunity? The generalization gives