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- The following theorem is part (i) of Theorem 7.11 of Kyprianou (2006) that gives necessary and sufficient conditions for a process to creep upwards or creep downwards. Theorem A.1. Suppose that X is a bounded variation L evy process which is not a compound Poisson process with the characteristic exponent Ψ(θ) := − log E eiθX1 . Then X creeps upwards (resp. downwards) if and only if the process X has the following L evyKhintchine exponent Ψ(θ) = −iθ + Z R−{0} (1 − eiθx ) v(dx), for > 0 (resp. < 0), and v is the L evy measure. Remark A.1. Since the process X in Hypothesis 2.1 is not a compound Poisson process and its drift is positive, Theorem A.1 is in force and the process X never creeps down.
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