Let (Z t ) t 0 be an R n -valued Lévy process. We consider stochastic differential equations of the formwhere Φ : R d → R d×n is Lipschitz continuous. We show that the infinitesimal generator of the solution process (X x t ) t 0 is a pseudo-differential operator whose symbol p :For a large class of Feller processes many properties of the sample paths can be derived by analysing the symbol. It turns out that the process (X x t ) t 0 is a Feller process if Φ is bounded and that the symbol is of the form p(x, ξ) = ψ(Φ ⊤ (x)ξ), where ψ is the characteristic exponent of the driving Lévy process. MSC 2010: 60J75; 47G30; 60H20; 60J25; 60G51; 60G17.
We propose new concepts in order to analyze and model the dependence structure between two time series. Our methods rely exclusively on the order structure of the data points. Hence, the methods are stable under monotone transformations of the time series and robust against small perturbations or measurement errors. Ordinal pattern dependence can be characterized by four parameters. We propose estimators for these parameters, and we calculate their asymptotic distributions. Furthermore, we derive a test for structural breaks within the dependence structure. All results are supplemented by simulation studies and empirical examples.For three consecutive data points attaining different values, there are six possibilities how their values can be ordered. These possibilities are called ordinal patterns. Our first idea is simply to count the number of coincidences of patterns in both time series, and to compare this with the expected number in the case of independence. If we detect a lot of coincident patterns, this means that the up-and-down behavior is similar. Hence, our concept can be seen as a way to measure non-linear 'correlation'. We show in the last section, how to generalize the concept in order to capture various other kinds of dependence. The authors gratefully acknowledge financial support of the DFG (German science Foundation) SFB 823: Statistical modeling of nonlinear dynamic processes (projects C3 and C5).
Comparison results are given for time-inhomogeneous Markov processes with respect to function classes induced stochastic orderings. The main result states comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbounded function classes is established. We give various applications to time-inhomogeneous diffusions, to processes with independent increments and to Lévy driven diffusion processes.
Let U be an open set in R d . We show that under a mild assumption on the richness of the generator a Feller process in (U, B(U )) with (predictable) killing is a semimartingale. To this end we generalize the notion of semimartingales in a natural way to those 'with killing'. Furthermore we calculate the semimartingale characteristics of the Feller process explicitly and analyze their connections to the symbol. Finally we derive a probabilistic formula to calculate the symbol of the process.MSC 2010: 60J75 (primary); 60J25, 60H05, 47G30, 60G51 (secondary).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.