A Ciesielski-Taylor type identity for positive self-similar Markov processes
Let \(Q^{(\nu)}\) be the law of the \(\nu\)-dimensional Bessel process starting at \(0\), where \(\nu>0\). Given a stochastic process \(\{X_t: t\geq 0\}\) and \(a>0\), let \(T_a=\inf\{s\geq 0: X_s=a\}\) and \(L_a=\int_{0}^{\infty} 1_{X_s\leq a}\,ds\). The Ciesielski-Taylor identity states that the law of \(T_a\) under \(Q^{(\nu)}\) is the same as the law of \(L_a\) under \(Q^{(\nu+2)}\). The aim of this paper is to provide a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type. The main result includes many previously known identities of the Ciesielski-Taylor type. Let \(\xi=\{\xi_t: t\geq 0\}\) be a spectrally negative Lévy process (with or without killing) with Laplace exponent \(\psi(u)=\log \operatorname{E} e^{u\xi_1}\). The authors show that \(\mathcal T_\beta\psi(u)=\frac{u}{u+\beta} \psi(u+\beta)\) is a Laplace exponent of another spectrally negative Lévy process for every \(\beta>0\). For a Lévy process \(\xi\) with Laplace exponent \(\psi\), denote by \(P_{\psi}\) the law of the positive self-similar Markov process starting at \(0\) which corresponds to \(\xi\) via the Lamperti transformation. The main result of the paper states that the law of \(T_a\) under \(P_{\psi}\) coincides with the law of \(L_a\) under \(P_{\mathcal T_{\alpha} \psi}\) for every \(\alpha>0\) such that \(\psi(\alpha)>0\). Moreover, the authors show that both random variables are self-decomposable and provide an explicit formula for their Laplace transform.
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