Skowroński, Arkadiusz On some properties of \(J\)-convex stochastic processes. (English) Zbl 0764.60040 Aequationes Math. 44, No. 2-3, 249-258 (1992). Some results on the differentiability of convex stochastic processes are presented. Next the following stochastic version of a theorem by Ng, Nikodem, Kominek on \(J\)-convex functions majorized by \(J\)-concave functions is given: Let \((\Omega,{\mathcal A},P)\) be a probability space and \((a,b)\subset\mathbb{R}\) be an open interval. If stochastic processes \(X_ 1,X_ 2: (a,b)\times\Omega\to\mathbb{R}\) are \(J\)-convex and \(J\)-concave, respectively, and for every \(t\in(a,b)\) satisfy the inequality \(X_ 1(t,\cdot)\leq X_ 2(t,\cdot)\) (a.e.), then there exist stochastic processes \(Y_ 1,Y_ 2:(a,b)\times\Omega\to\mathbb{R}\) and \(A:(a,b)\times\Omega\to\mathbb{R}\) such that \(Y_ 1\) is convex, \(Y_ 2\) is concave, \(A\) is additive and \(X_ 1=A+Y_ 1\) and \(X_ 2=A+Y_ 2\). Reviewer: A.Skowroński (Bielsko-Biała) Cited in 27 Documents MSC: 60G07 General theory of stochastic processes 26A51 Convexity of real functions in one variable, generalizations 39B72 Systems of functional equations and inequalities 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 60G99 Stochastic processes 60H99 Stochastic analysis 60K99 Special processes Keywords:differentiability of convex stochastic processes PDFBibTeX XMLCite \textit{A. Skowroński}, Aequationes Math. 44, No. 2--3, 249--258 (1992; Zbl 0764.60040) Full Text: DOI EuDML References: [1] Billingsley, P.,Probability and measure. Wiley, New York–Chichester–Brisbane, 1979. · Zbl 0411.60001 [2] Kominek, Z.,On a problem of K. Nikodem. Arch. Math.50 (1988), 287–288. · Zbl 0642.39005 · doi:10.1007/BF01187747 [3] Nagy, B.,On a generalization of the Cauchy equation. Aequationes Math.10 (1974), 165–171. · Zbl 0294.60052 · doi:10.1007/BF01832853 [4] Ng, C. T.,On midconvex functions with midconcave bounds. Proc. Amer. Math. Soc.102 (1988), 538–540. · Zbl 0659.39004 · doi:10.1090/S0002-9939-1988-0928975-7 [5] Nikodem, K.,Midpoint convex functions majorized by midpoint concave functions. Aequationes Math.32 (1987), 45–51. · Zbl 0612.26007 · doi:10.1007/BF02311298 [6] Nikodem, K.,On convex stochastic processes. Aequationes Math.20 (1980), 184–197. · Zbl 0443.60032 · doi:10.1007/BF02190513 [7] Nikodem, K.,Wypukle i kwadratowe procesy stochastyczne. Thesis, Silesian University, Katowice, 1980, (doctoral dissertation). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.