The sharp Markov property of Lévy sheets
This article extends the work of the authors on the strong Markov property (SMP) of the Brownian sheet [Acta Math. 168, No. 3/4, 153-218 (1992; Zbl 0759.60056)] to two-parameter Lévy processes \(X\). They decompose \(X\) into a Gaussian and a jump part and show that the second part satisfies the SMP with respect to all open sets which are bounded or have bounded complement, where the sharp field (i.e. the field generated by \(X)\) has to be replaced by the uniform sharp field if the sign of the jumps is not constant. This result extends the SMP of the Poisson sheet proved by \textit{E. Carnal} and \textit{J. B. Walsh} [Stochastic analysis, Proc. Conf. Honor Moske Zakai 65th Birthday, Haifa/Isr. 1991, 91-110 (1991; Zbl 0734.60052)] and shows clearly that the SMP is a less restrictive property for jump processes than for continuous processes. Moreover the authors prove that the SMP of \(X\) implies the SMP of both components in the decomposition. They also consider the minimal splitting field.
scientific article; zbMATH DE number 89777