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We derive tail asymptotics for the running maximum of the Cox-Ingersoll-Rossprocess. The main result is proved by the saddle point method, where the tailestimate uses a new monotonicity property of the Kummer function. Thisauxiliary result is established by a computer algebra assisted proof. Moreover,we analyse the coefficients of the eigenfunction expansion of the runningmaximum distribution asymptotically.

The running maximum of the Cox-Ingersoll-Ross process with some properties of the Kummer function