Vol. 58, No. 2, 1975

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The inverse of a continuous additive functional

Jean-Marie G. Rolin

Vol. 58 (1975), No. 2, 585–604

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 = Xτ. 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.

Mathematical Subject Classification 2000
Primary: 60J55
Received: 22 March 1974
Published: 1 June 1975
Jean-Marie G. Rolin