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On some properties of \(J\)-convex stochastic processes. (English) Zbl 0764.60040

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 \(X_ 2=A+Y_ 2\).

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
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References:

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