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Perturbation analysis of Poisson processes. (English) Zbl 1303.60039
Let \(\Phi\) be a Poisson process on a measurable space \((\mathbb{X}, \mathcal{X})\). Denote by \(\Pi_{\lambda}\) the distribution of a Poisson process with intensity measure \(\lambda\) (\(\lambda\) is a \(\sigma\)-finite measure on \((\mathbb{X}, \mathcal{X})\)). Let \(f(\Phi)\) be some measurable function of \(\Phi\). Under certain assumptions on \(f\), the authors of [I. Molchanov and S. Zuyev, Math. Oper. Res. 25, No. 3, 485–508 (2000; Zbl 1018.49022)] derived a variational formula for the finite measure \(\lambda\). In the present paper, the author extends the variational formula to the \(\sigma\)-finite measure \(\lambda\). The extension of the identity from finite to \(\sigma\)-finite measures is a non-trivial task. The approach is based on the recent Fock space representation in [G. Last and M. D. Penrose, Probab. Theory Relat. Fields 150, No. 3–4, 663–690 (2011; Zbl 1233.60026)].
The paper is organized as follows. In Section 2, the author introduces some basic notation and recalls facts about the Fock space representation and likelihood functions of Poisson processes. Section 3 uses an elementary but illustrative argument to prove a simple version of the variational formula from [Molchanov and Zuyev, loc. cit.]. In Section 4, the author proves the main result of the paper. Section 5 derives conditions on \(\lambda\) that are necessary for the variational identity to hold for all bounded functions \(f\). In some cases, these conditions are also sufficient. Section 6 gives general Margulis-Russo-type formulas for derivatives. The final Section 7 treats perturbations of the Lévy measure of a Lévy process in \({\mathbb{R}}^d\).

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G51 Processes with independent increments; Lévy processes
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