Modeling of Continuous Time Index Fund and its Statistical Test

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This research is concerned with a continuous time financial problem to find an index fund similar to the benchmark portfolio. The problem is formulated into the optimal stochastic control problem under the mean square error tracking at the final point in time horizon. The main results obtained in this research are some extensions of the static case and described as follows:(1) If the benchmark portfolio is on the efficient frontier, then the optimal index fund is also on the efficient frontier for any point in time.(2) If the benchmark portfolio is not on the efficient frontier, then the optimal index fund is not also on the efficient frontier for any point in time.(3) The genetic algorithm is most efficient to find an optimal index fund.One of our results is published in the discussion paper and the comments and reflects from researchers have been required. The other paper is submitted to the academic journal. Moreover, all the results were reported in the workshop in the OR society of Japan and in the international conference held in Korea. In addition, two books in which include the fundamental concept of our results are published in Japan.