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Unbiased shifts of Brownian motion. (English) Zbl 1295.60093

Consider a two-sided standard Brownian motion \(B= (B_t)_{t\in\mathbb{R}}\) and a probability measure \(\nu\) on \(\mathbb{R}\). Among the random, not necessarily stopping, times \(T\) which solve the Skorodhod embedding problem: the law of \(B_T\) is \(\nu\), there are the so-called “unbiased shifts”, i.e., those \(T\) which are \(B\)-measurable and such that \((B_{T+t}- B_T)_{t\in\mathbb{R}}\) is Brownian too and independent of \(B_T\).
This article focuses on those unbiased shifts. First, the authors characterize all of them, on the one hand in terms of the allocation rule \(t\mapsto t+T_0\theta_t\) and its relationship to local times, and on the other hand in terms of Palm measures associated with local times. The authors then take advantage of these characterizations to construct unbiased shifts, and in particular a solution which is also a stopping time, using a stopping time previously introduced by J. Bertoin and Y. Le Jan [Ann. Probab. 20, No. 1, 538–548 (1992; Zbl 0749.60038)].
Moreover, integrability and minimality properties of the solutions are discussed, and finally, more general Lévy processes are also considered. Links are made with the so-called matching of Poisson processes, and the whole article is very well written.

MSC:

60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 0749.60038
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References:

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