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Erschienen in:
15.03.2022
Martingales associated with functions of Markov and finite variation processes
Erschienen in: Queueing Systems | Ausgabe 3-4/2022
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Excerpt
Suppose that \(X=\{X_t|\ t\ge 0\}\) is a càdlàg Markov process taking values in some metric space (where the notion of càdlàg is well defined) with respect to some filtration satisfying the usual conditions and that \(Y=\{Y_t|\ t\ge 0\}\) is a \(\mathbb {R}^r\)-valued càdlàg, adapted process of finite variation on finite intervals. Assume that X has an (extended) generator \(\mathscr {A}\) such that for any continuous \(\xi \) in its domain (so that \(\xi (X_t)\) is also càdlàg and adapted), we have that
$$\begin{aligned} \xi (X_t)-\xi (X_0)-\int _0^t\mathscr {A}\xi (X_s)ds \end{aligned}$$
(1)
$$\begin{aligned} \eta (Y_t)=\eta (Y_0)+\int _0^t\nabla \eta (Y_s)^TdY^c_s+\sum _{0<s\le t}\varDelta \eta (Y_s) \end{aligned}$$
(2)