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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 $$\frac{p}{2}$$.

##### MSC:
 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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