Schnurr, Alexander
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Publication Access status: Metadata only , From Markov processes to semimartingalesIn the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the 'natural' class of processes for many concepts first developed in the Markovian framework. As an example, stochastic differential equations have been invented as a tool to study Markov processes but nowadays are treated separately in the literature. Moreover, the killing of processes has been known for decades before it made its way to the theory of semimartingales most recently. We describe, when these and other important concepts have been invented in the theory of Markov processes and how they where transferred to semimartingales. Further topics include the symbol, characteristics and generalizations of Blumenthal-Getoor indices. Some additional comments on relations between Markov processes and semimartingales round out the paper.Source Type:ArticleDOI:10.1214/23-ps193 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , The Bayesian Group-Sequential Predictive Evidence Value Design for Phase II Clinical Trials with Binary Endpoints(2024); In clinical research, the initial efficacy of a new agent is typically assessed in a phase IIA study. Bayesian group-sequential designs are often based on predictive probability of trial success. In this paper, the novel Bayesian group-sequential predictive evidence value design is introduced, and we prove that the predictive probability approach is a special case of it. A comparison with Simon’s two-stage and competing Bayesian designs based on phase IIA cancer trials is provided. Results show that the novel design can improve operating characteristics such as the false-positive rate, probability of early stopping for futility and expected sample size of the trial. Given these advantages, the predictive evidence value design constitutes an important addition to the biostatistician’s toolbelt when planning a phase IIA trial the Bayesian way, in particular, when small sample sizes and a large probability for early termination under the null hypothesis are desired.Source Type:Article1 - Some of the metrics are blocked by yourconsent settings
Publication Access status: Metadata only , The Time-Dependent Symbol of a Non-homogeneous Itô Process and Corresponding Maximal InequalitiesThe probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present Article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p -variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.Source Type:Article3
