The class of distributions of periodic Ornstein-Uhlenbeck processes driven by Lévy processes
Let \(Z\) be a Lévy process on \([0,1]\) with values in \(\mathbb R^d\) and \(c\in\mathbb R\setminus\{0\}\). Then the unique solution \(X\) to the Langevin equation \(dX_{t}=-cX_{t}dt + dZ_{t} \) for \(t\in[0,1]\) and \(X_{0}=X_{1}\) a.s. is called periodic Ornstein-Uhlenbeck process. Its canonic pathwise periodic extension on the real axis is a stationary process and in particular its stationary distribution satisfies \({\mathcal L}(X_{t})={\mathcal L}(X_{1})={\mathcal L}((e^c-1)^{-1}\int_{0}^{1}e^{cs}dZ_{s})\) on \([0,1]\). The authors are interested in the analysis of the class \(I(c)\) of stationary distributions of periodic Ornstein-Uhlenbeck processes with parameter \(c\) where the driving Lévy processes are ranging over all Lévy processes on \(d\)-dimensional Euclidean space. A usual Ornstein-Uhlenbeck process \(Y\) with parameter \(c>0\) on \([0,\infty)\) satisfies \({\mathcal L}(Y_{t})={\mathcal L}(Y_{0})\) for \(t\in[0,\infty)\) if and only if \({\mathcal L}(Z_{1})\) has finite \(\log\)-moment and \({\mathcal L}(Y_{0})={\mathcal L}(\int_{0}^{1}e^{-cs}dZ_{s})\). In this case the stationary distribution \({\mathcal L}(Y_{0})\) is selfdecomposable. Conversely, a selfdecomposable distribution \(\mu\) can be expressed in the same way with a Lévy process having finite \(\log\)-moment. The correspondence of \(\mu\) to \({\mathcal L}(Z_{1})\) is one to one for fixed \(c>0\). Thus the class of stationary distributions of usual Ornstein-Uhlenbeck processes does not depend on \(c\) and coincides with the class SD\((\mathbb{R}^d)\) of selfdecomposable distributions on \(d\)-dimensional Euclidian space.. For \(I(c)\) on the contrary the authors show that this class delicately depends on the parameter \(c\). They characterize \(I(c)\) in terms of properties of the Lévy measures of the distributions in this class. The mapping \(\varphi_{c}\) of \(\mu_{0}\) to \(\mu=\varphi_{c}(\mu_{0})={\mathcal L}(\int_{0}^{1}e^{cs}dZ_{s})\) is shown to be bijective and homeomorphic. Moreover, the relations in between the classes \(I(c)\) for varying values of \(c\) and with SD\((\mathbb{R}^d)\) are studied. It is proved that SD\((\mathbb{R}^d)\) is a real subset of \(I(c)\) and \(\bigcap_{n=1}^{\infty}I(c_{n})=\text{SD}(\mathbb{R}^d)\) whenever \(c_{n}\) converges to \(\infty\). One way to look at the dependence of \(I(c)\) on \(c\) is to study, given \(\mu\) in the class of infinitely divisible distributions, the set \({\mathcal Q}(\mu)=\{c\geq 0:\mu\in I(c)\}\) and showing that these sets have a rich variety inferring over \(\mu\). Finally the relations of \(I(c)\) with the nested classes \(L_{m}\) are discussed.
scientific article; zbMATH DE number 2170060
scientific article