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OPTIMAL STOPPING IN THE Llog L-INEQUALITY OF HARDY AND LITTLEWOOD

Published online by Cambridge University Press:  01 March 1998

S. E. GRAVERSEN
Affiliation:
Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark
G. PEšKIR
Affiliation:
Institute of Mathematics, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark Department of Mathematics, University of Zagreb, Bijenička 30, 41000 Zagreb, Croatia
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Abstract

Let B=(Bt)t[ges ]0 be standard Brownian motion started at zero. We prove

formula here

for all c>1 and all stopping times τ for B satisfying E(τr)<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

τ*=inf{t>0[mid ]St [ges ]u*, Xt =1∨αSt},

where u*=1+1/ec(c−1) and α=(c−1)/c for c>1, with Xt=[mid ]Bt[mid ] and St= max0[les ]r[les ]t[mid ]Br[mid ]. Likewise, we prove

formula here

for all c>1 and all stopping times τ for B satisfying Er<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

σ*=inf{t>0[mid ]St [ges ]v*, XtSt},

where v*=c/e(c−1) and α=(c−1)/c for c>1. These results contain and refine the results on the Llog L-inequality of Gilat [6] which are obtained by analytic methods. The method of proof used here is probabilistic and is based upon solving the optimal stopping problem with the payoff

formula here

where F(x) equals either xlog+x or xlog x. This optimal stopping problem has some new interesting features, but in essence is solved by applying the principle of smooth fit and the maximality principle. The results extend to the case when B starts at any given point (as well as to all non-negative submartingales).

Type
Research Article
Copyright
© The London Mathematical Society 1998

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