Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations

Relating distributed, temporal and causal observations of simple processes

The noninterleaving equivalences that have been proposed in the literature may be roughly divided into three classes, depending on the kind of observational scenarios underlying them. Some of them distinguish parallelism from nondeterminism by admitting explicit observations of the causal structure of systems; some others are based on the observation of distribution and still others on the assu...

Bandit Problems with Lévy Processes

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Lévy processes and continuous - state branching processes : part I

Let us begin by recalling the definition of two familiar processes, a Brownian motion and a Poisson process. A real-valued process B = {B t : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a Brownian motion if the following hold: (i) The paths of B are P-almost surely continuous. (ii) P(B 0 = 0) = 1. (iii) For 0 ≤ s ≤ t, B t − B s is equal in distribution to B t−s. (iv) For 0 ≤ ...

Girsanov identities for Poisson measures under quasi-nilpotent transformations

Sequential Testing Problems for Poisson Processes

We present the explicit solution of the Bayesian problem of sequential testing of two simple hypotheses about the intensity of an observed Poisson process. The method of proof consists of reducing the initial problem to a free-boundary differential-difference Stephan problem, and solving the latter by use of the principles of smooth and continuous fit. A rigorous proof of the optimality of the ...