Lower and Upper Bounds for the Discrete Bi-Directional Preemptive Conversion Problem with a Constant Price Interval

Abstract

In the conversion problem, wealth has to be distributed between two assets with the objective to maximize the wealth at the end of the investment horizon. The bi-directional preemptive conversion problem with a constant price interval is the only problem, of the four main variants of the conversion problem, that has not yet been optimally solved by competitive analysis. Assuming a given number of monotonous price trends called runs, lower and upper bounds for the competitive ratio are given. In this work, the assumption of a given number of runs is rejected and lower and upper bounds for the bi-directional preemptive conversion problem with a constant price interval are given. Furthermore, an algorithm based on error balancing is given which at minimum achieves the given upper bound. It can also be shown that this algorithm is optimal for the single-period model.