Numerical approaches to finite-horizon optimal stopping problems for the Shiryaev process

In a finite-horizon optimal stopping problem the optimal stopping time is typically given by the first moment at which a sufficient statistic, namely a process containing all the relevant information on the problem, exceeds an unknown time-dependent boundary. 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. We demonstrate that these approaches return accurate results and are generally efficient.