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.
Highlights
In the conversion problem (CP), an initial wealth W0 is distributed between two assets, e.g., dollars and yen
Authors in [19] used EBA to give an upper bound for the bi-directional preemptive conversion problem with constant price interval and unknown k
A lower upper bound for the bi-directional preemptive CP with a constant price interval can be found by determining the competitive ratio of EBA for T > 3
Summary
In the conversion problem (CP), an initial wealth W0 is distributed between two assets, e.g., dollars and yen. Reference [8] considers the uni-directional non-preemptive CP and provides the online algorithm RPP, which solves the variant of the CP optimally, in the case that M and m are given. CP with given M, m, and T is considered by [3,9] They provide the optimal online algorithm uTH. Researchers in [12] provide (among other things) an optimal algorithm for the bi-directional preemptive CP with interrelated prices. Authors in [14] considered the uni-directional preemptive CP, but they divided the wealth into u equal valued units and it is only allowed to convert one unit per point in time t. The authors in [15] considered a problem that is allowed to convert an integer number of units per point in time. These bounds hold for a given T, discrete time, and a constant price interval between the price bounds m and M
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