Let X be a standard
process and A be a continuous additive functional of X. The inverse of A
is defined by τt=inf{s ⋅ As> t}. The aim of this paper is to prove that
the process τ has conditionally independent increments with respect to the
σ-algebra generated by the time changed process Xl= Xτℓ. However these
increments are not necessarily stationary. Another interesting result is derived: the
continous part of the process τ is a continuous additive functional of the process
X.
The existence of regular conditional probabilities permits to consider the process
τ as an additive process and under a necessary and sufficient condition, it is in fact a
Levy process with increasing paths. The general theory of such processes
is then used to obtain a Levy representation of the iumps of the process
τ.
