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This boundary often turns out to be the solution of a highly nonlinear integral equation involving the transition density of the sufficient statistic. When this density cannot be computed directly or easily, standard methods for solving the integral equation must be modified. This situation arises in sequential detection problems and in the pricing of certain derivative securities, where the corresponding sufficient statistics follow the so called Shiryaev process. In this context, we analyze and implement three distinct numerical methods for solving the integral equations characterizing the associated optimal stopping boundaries: two of them rely on solutions to partial differential equations, while the third is based on approximating the distribution of the sufficient statistic using a log-normal distribution. 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