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Remarks on Gross’ technique for obtaining a conformal Skorohod embedding of planar Brownian motion. (English) Zbl 1434.60224
Summary: In [R. Gross, Electron. Commun. Probab. 24, Paper No. 68, 11 p. (2019; Zbl 1450.60032)] it was proved that, 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 \(\mathcal{R} e(Z_{\tau_{\Omega}})\) has the distribution \(\mu \), where \(\tau_{\Omega}\) denotes the exit time of \(Z_t\) from \(\Omega \). In this note, we extend this method 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 \(\frac{p} {2} \). We also prove a uniqueness principle for this construction, and use it to give several examples.

MSC:
60J65 Brownian motion
30C20 Conformal mappings of special domains
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