Logarithmic Estimates for Submartingales and Their Differential Subordinates

Adam Osękowski ()
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Adam Osękowski: University of Warsaw

Journal of Theoretical Probability, 2011, vol. 24, issue 3, 849-874

Abstract: Abstract In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then $$\sup_t\mathbb{E}|Y_t|\leq K\sup_t\mathbb{E}X_t\log^+X_t+L(K,\alpha).$$ Related sharp inequalities for martingales are also established. As an application, we obtain logarithmic estimates for smooth functions on Euclidean domains.

Keywords: Martingale; Submartingale; Subharmonic function; Maximal function; Differential subordination; Strong differential subordination; LlogL class; Boundary value problem; 60G42; 31B05 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-009-0268-9

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