On some properties of \(J\)-convex stochastic processes
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 \(X_ 2=A+Y_ 2\).
differentiability of convex stochastic processes
scientific article; zbMATH DE number 149475