On the distribution of the Hilbert transform of the local time of a symmetric Lévy process
Let \((X_ t,P^ x)\) be a symmetric, real-valued Lévy process \((P^ x\) is the probability law under the stipulation that \(X_ 0\equiv x)\), with \(E^ 0[e^{i\lambda X_ t}]=e^{-t\psi(\lambda)}\), where \(\psi(\lambda)={1\over 2}\sigma^ 2\lambda^ 2+\int^ \infty_ 0(1- \cos\lambda x)dx\) satisfies \(\kappa(q){\buildrel{\text{def}} \over =}\int^ \infty_ 0{d\lambda\over q+\psi(\lambda)}<\infty\) for all \(q>0\). Assume that there exists a jointly continuous version \((L^ x_ t)\) of the local time for the process, and define \[ H_ t={1\over\pi}\int^ \infty_ 0{1\over y}(L^ y_ t-L_ t^{-y})dt. \] Finally, let \[ g(t)=\sup\{s\leq t:X_ s=0\},\quad d(t)=\inf\{s>t:X_ s=0\}. \] With \(q\) fixed, let \(T\) be exponentially distributed and independent of \(X\). The main results are computational formulas for the random variables \(H_ T\), \(H_{g(T)}\), \(H_ T-H_{g(T)}\), \(H_{d(T)}\) and \(H_{d(T)}-H_{g(T)}\). For example, \[ E^ 0[e^{i\lambda H_ T}]=\text{sech}(\lambda\kappa(q)),\quad E^ 0[e^{i\lambda H_{g(T)}}]={\tanh(\lambda\kappa(q))\over\lambda\kappa(q)}, \] \[ E^ 0[e^{i\lambda H_{g(T)}+i\mu[H_ t- H_{g(T)}]}]={\tanh(\lambda\kappa(q))\over\lambda} {\mu\over\sinh(\mu\kappa(q))}. \] In particular, \(H_ T\) and \(H_ T- H_{g(T)}\) are independent. These results generalize work of \textit{Ph. Biane} and \textit{M. Yor} [Bull. Sci. Math., II. Sér. 111, 23-101 (1987; Zbl 0619.60072)] for Brownian motion. The proofs are based on moment calculations.
scientific article; zbMATH DE number 97406