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

Abstract

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

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

τ_*=inf{t>0[mid ]S_t [ges ]u_*, X_t =1∨αS_t},

where u_*=1+1/e^c(c−1) and α=(c−1)/c for c>1, with X_t=[mid ]B_t[mid ] and S_t= max_{0[les ]r[les ]t}[mid ]B_r[mid ]. Likewise, we prove

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

σ_*=inf{t>0[mid ]S_t [ges ]v_*, X_t =αS_t},

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

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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).