XXIX: 16, 166-180, LNM 1613 (1995)
APPLEBAUM, David
A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)
This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.
Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (Stochastics Stochastics Rep. 56, 1996). The same question is addressed by Cohen in the next article 2917
Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators
Nature: Original
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XXIX: 17, 181-193, LNM 1613 (1995)
COHEN, Serge
Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)
The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (Stochastics Stochastics Rep. 56, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind
Comment: The first definition is independently introduced by David Applebaum 2916
Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators
Nature: Original
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