Summary
SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt 0 and δ>0 such thatZ(t 0) is an extremal point of the convex hull of {Z(t)|t 0−δ≦t≦t0} and also an extremal point of the convex hull of {Z(t)|t 0≦t≦t0+δ} and, moreover, the tangent lines to the convex hulls atZ(t 0) form a non-zero angle.
The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t 0 and δ>0 such that the intersection of the convex hulls of {Z(t)|t 0−δ≦t≦t0} and {Z(t)|t 0≦t≦t0+δ} contains onlyZ(t 0)?
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This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture
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Shimura, M. Flat points of two-dimensional Brownian motion. Probab. Th. Rel. Fields 93, 197–229 (1992). https://doi.org/10.1007/BF01195229
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DOI: https://doi.org/10.1007/BF01195229
Keywords
- Hull
- Stochastic Process
- Brownian Motion
- Probability Theory
- Convex Hull