A collocation method for the sequential testing of a gamma process

We study the Bayesian problem of sequential testing of two simple hypotheses about the parameter α > 0 of a Lévy gamma process. The initial optimal stopping problem is reduced to a free-boundary problem where, at the unknown boundary points separating the stopping and continuation set, the principles of the smooth and/or continuous fit hold and the unknown value function satisfies on the continuation set a linear integro-differential equation. Due to the form of the Lévy measure of a gamma process, determining the solution of this equation and the boundaries is not an easy task. Hence, instead of solving the problem analytically, we use a collocation technique: the value function is replaced by a truncated series of polynomials with unknown coefficients that, together with the boundary points, are determined by forcing the series to satisfy the boundary conditions and, at fixed points, the integro-differential equation. The proposed numerical technique is employed in well-understood problems to assess its efficiency.