Erratum - Some Characterizations of Mixed Poisson Processes

Abstract

Assumption A. Let X := {Xt}t∈R+ be a process with right-continuous paths such that its canonical filtration F := {Ft}t∈R+ is also right-continuous. Sketch of the proof of Lemma 4.1. For the “only if” part, assuming that X is P -conditionally independent, it follows that there exists a PΘ-null set N ∈ B such that, for any θ / ∈ N , {Xt}t∈Q+ is Pθindependent, i.e., the condition (∗) : EPθ [χAf(Xt)] = Pθ(A)EPθ [f(Xt)] is true, for all s, t ∈ Q+ with s < t, all bounded Borel measurable f : R → R and all A ∈ Fs. Using standard arguments, condition (∗) can be proved for any s, t ∈ R+ with s < t. In Lemma 4.6, put F := { Ft}t∈R+ , assume for (i) =⇒ (ii) as well as for Proposition 4.8, thatX is non-decreasing, and insert assertion “In particular, if Θ ∈ L1(P ) then {Nt − tΘ}t∈R+ is a (P, A)-martingale if and only if {Nt − tθ}t∈R+ is a (Pθ, A)-martingale for PΘ-a.a. θ ∈ R”. In Theorem 4.10, the families {Nt − EPθ [Nt]}t∈R+ and {Nt − EP [Nt | Θ]}t∈R+ must be replaced by {Nt − tθ}t∈R+ and {Nt − tΘ}t∈R+ respectively, with the proof remaining unchanged.