Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction |
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Authors: | Chris Carolan Richard Dykstra |
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Affiliation: | (1) Department of Mathematics, East Carolina University, 129 Austin, 27858 Greenville, NC, U.S.A.;(2) Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer, 52242 Iowa City, IA, U.S.A. |
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Abstract: | The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented. |
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Keywords: | Brownian motion least concave majorant |
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