On the approximation by the continuous time models

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The object of this project is to carry out an experiment on the ability to apply the optimal sequential test for determining the sign of a normal mean as an approximation to the solution of a large class of sequential testing problems. Two potential advantages of such an approximation, if it is proved to be effective, are that it is easy to calculate and that it can be expressed simply in terms of a single easily interpreted curve, determine the optimal stopping time. Incidentally one version of that curve is one where a normal significance level for stopping is expressed as a function of the information obtained to date.The experiment consists of comparing the approximation to the optimal Bayes solution for the sequential test that the standard deviation of a normal distribution exceeds one. In this problem the unknown value of θ = σ-2 is assumed to have a gamma distribution. The naive application of the central limit theorem gives us unsatisfactory approximations. The Edgeworth type approximation has been considered together with the continuous time approximation for the discrete time processes. The numerically satisfactory results have not been obtained yet. The constant proposed by Hogan (1986, Ann. Statists.) is obtained for the gamma random walk.Another problem considered in this project is the pricing of the exponentially square root curved boundary knockout type European type option. We have obtained the pricing formula, a part of which has been published (Morimoto, M 1999, Ph.D. thesis Boston University).