Optimal Stopping in the L log L-Inequality of Hardy and Littlewood |
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Authors: | Graversen, S. E. Peskir, G. |
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Affiliation: | Institute of Mathematics, University of Aarhus Ny Munkegade, 8000 Aarhus, Denmark Institute of Mathematics, University of Aarhus Ny Munkegade, 8000 Aarhus, Denmark and Department of Mathematics, University of Zagreb Bijenika 30, 41000 Zagreb, Croatia |
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Abstract: | Let B = (Bt)t0 be standard Brownian motion started at zero.We prove for all c > 1and all stopping times for B satisfying E(r) < for somer > 1/2. This inequality is sharp, and equality is attainedat the stopping time whereu* = 1 + 1/ec(c 1) and = (c 1)/c for c >1, with Xt = |Bt| and St = max0rt|Br|. Likewise, we prove for all c > 1 and all stopping times for B satisfying E(r < for some r > 1/2. This inequalityis sharp, and equality is attained at the stopping time where v* = c/e(c 1) and =(c 1)/c for c > 1. These results contain and refinethe results on the L log L-inequality of Gilat [6] which areobtained by analytic methods. The method of proof used hereis probabilistic and is based upon solving the optimal stoppingproblem with the payoff whereF(x) equals either xlog+ x or x log x. This optimal stoppingproblem has some new interesting features, but in essence issolved by applying the principle of smooth fit and the maximalityprinciple. The results extend to the case when B starts at anygiven point (as well as to all non-negative submartingales).1991 Mathematics Subject Classification 60G40, 60J65, 60E15. |
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