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In this Note we study optimal stopping problems for strong Markov processesand affine functions. We give a justification of the Snell envelope form usingstandard results of optimal stopping. We also justify the convexity of thevalue function, and without a priori restriction to a particular class ofstopping times, we deduce that the smallest optimal stopping time isnecessarily a hitting time.

Arr\^et optimal pour les processus de Markov forts et les fonctions affines