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A conformal Skorokhod embedding. (English) Zbl 1450.60032
In this paper, the author gives a construction of the following realisation of the random variable with given distribution. Start a planar Brownian motion and let it run until it hits some given barrier. The barrier may be crafted so that the \(x\) coordinate at the hitting time has any prescribed centered distribution with finite variance. Roughly speaking, given a distribution \(\mu\) with zero mean and finite second moment, there exists a simply connected domain \(\Omega\) such that if \(Z_t\) is a standard planar Brownian motion, then \(\mathrm{Re}(Z_{\tau_\Omega})\) has the distribution \(\mu\), where \(\tau_\Omega\) denotes the exit time of \(Z_t\) from \(\Omega\). This provides a new, complex-analytic proof of the Skorokhod embedding theorem. The method is constructive, i.e., it can give an explicit description of the barrier.
Recently, in [M. Boudabra and G. Markowsky, Electron. Commun. Probab. 25, Paper No. 20, 13 p. (2020; Zbl 1434.60224)] the result has been extended to prove that if \(\mu\) has a finite \(p\)-th moment then the first exit time \(\tau_{\Omega}\) from \(\Omega\) has a finite moment of order \(\frac{p}{2}\).

60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
30B10 Power series (including lacunary series) in one complex variable
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