On Jensen’s inequality, Hölder’s inequality, and Minkowski’s inequality for dynamically consistent nonlinear evaluations

摘要

In this paper, the dynamically consistent nonlinear evaluations that were introduced by Peng are considered in probability space L 2 ( Ω , F , ( F t ) t ≥ 0 , P ) $L^{2} (Omega,{mathcal{F}}, ({mathcal {F}}_{t} )_{tgeq0},P )$ . We investigate the n-dimensional ( n ≥ 1 $ngeq1$ ) Jensen inequality, Hölder inequality, and Minkowski inequality for dynamically consistent nonlinear evaluations in L 1 ( Ω , F , ( F t ) t ≥ 0 , P ) $L^{1} (Omega,{mathcal{F}}, ({mathcal {F}}_{t} )_{tgeq0},P )$ . Furthermore, we give four equivalent conditions on the n-dimensional Jensen inequality for g-evaluations induced by backward stochastic differential equations with non-uniform Lipschitz coefficients in L p ( Ω , F , ( F t ) 0 ≤ t ≤ T , P ) $L^{p} (Omega,{mathcal{F}}, ({mathcal {F}}_{t} )_{0leq tleq T},P )$ ( 1 p ≤ 2 $1 pleq2$ ). Finally, we give a sufficient con