応用数理

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A lot of models are included in the optimal stopping problem. Here, we treat the job search problem as a clue of the model to maximize the expected value. In addition, it thinks about a sequential stochastic assignment problem and a job search problem on a partially observable Markov chain. The job search is a problem for a sequence of the random variables appearing one by one, to select one from among them, and maximizing the expected value. It is possible to extend variously, and it explains a sequential stochastic assignment problem as the one. On the other hand, though it is not possible to know the state of the Markov chain directly, information is obtained through the information process, this process is said the partially observable Markov chain. It thinks about the Bayesian learning procedure for this chain. The order is introduced into the information space of use the likelihood ratio, and make a preparation to analyze the property of the sequential decision problems in this Markov chain and necessary properties are brought together for that. Finally, it thinks about the job search in this Markov chain as a sequential decision problem of incomplete information.

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A problem of optimal stopping of the discrete time Markov process by two decision makers in a competitive situation is considered. We briefly follow Dynkin Game and discuss its extension to a multiple stopping model. The double stopping problem is recalled and then a stopping game with priority for Player 1 and the two stops for Player 2 will be formulated. By using the solution of an auxiliary double stopping problem, our aim is to construct the equilibrium pair in the game. The examples of such decision problems is related to variation of the best choice problem and also is a generalization of the other stopping games.

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Under the no-arbitrage condition, the pricing (or evaluation of the fair price) of an American derivative is reduced to an optimal stopping problem under a risk neutral (or an equivalent martingale) probability measure, which involves various interesting problems related to several fields of mathematical sciences. In this article, first, we briefly explain the main principles of this risk neutral evaluation method in the framework of Black-Scholes (-Merton) market. And, then, we present several classical and recently derived results concerning the analytical characterizations of the optimal value function and optimal stopping time, which are useful for its both analytical and numerical evaluation.

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It seems to be generally agreed among researchers that the Martin Gardner's Mathematical Games column in the February 1960 issue of Scientific American is the first statement of the secretary problem. Since then the problem has been extended and generalized in many different directions. The object of this article is to show how extensive and vast the field has grown and review the recent works of interest, especially published in the last decade. For the previous works, the reader is referred to review papers by Ferguson (1989) or Samuels (1991).

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