We study the Bayesian problem of sequential testing of two simple hypotheses about the parameter α &gt; 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.

Buonaguidi, B., Muliere, P., A collocation method for the sequential testing of a gamma process, <<STATISTICA SINICA>>, 2015; 25 (4): 1527-1546. [doi:10.5705/ss.2013.155] [http://hdl.handle.net/10807/133701]

### A collocation method for the sequential testing of a gamma process

#### Abstract

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.
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Buonaguidi, B., Muliere, P., A collocation method for the sequential testing of a gamma process, <<STATISTICA SINICA>>, 2015; 25 (4): 1527-1546. [doi:10.5705/ss.2013.155] [http://hdl.handle.net/10807/133701]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10807/133701`
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