Optimal Stopping and Switching in a Matrix of Random Variables and Some Related Inequalities

摘要

Let X be a matrix of integrable random variables. Each row of the matrix represents the value of an object fluctuating in time. A speculator has the following options to trade with these objects: he is allowed to possess one object at a time, he can sell an object and buy a new one at any time, and he can stop with speculating whenever he wants. The speculator has to pay a fraction of the value of an object when he decides to sell that particular object. Also, the selling strategy may depend on past outcomes. The reward of the speculator is defined as the supremum over all strategies and stopping times of the process determined by the strategy and stopped at time. In the report an algorithm is given to construct the best strategy in the case that X has finite number of columns. A comparison is given between the reward of X defined here and the optimal stopping reward of Hill and Kennedy. In sections 4 and 5 some so-called prophet inequalities are given. In particular, the prophet inequality of Hill and Kennedy is proved alternatively.