Remarks on strongly convex stochastic processes (Q372380)

Let \((\Omega, {\mathcal A}, \operatorname P)\) be an arbitrary probability space and \(C:\Omega \rightarrow \mathbb{R}\) denote a positive random variable. We say that a stochastic process \(X: I\times \Omega \rightarrow \mathbb{R}\) is strongly convex with modulus \(C(\cdot )>0\) if the inequality \[ X(\lambda u +(1-\lambda)v, \cdot ) \leq \lambda X(u,\cdot ) +(1-\lambda)X(v,\cdot)-C(\cdot)\lambda (1-\lambda)(u-v)^2 \qquad \text{(a.e.)} \] is satisfied for all \(\lambda \in [0,1]\) and \(u,v \in I\). If the above inequality is assumed only for \(\lambda =\frac 12\), then the process \(X\) is called strongly Jensen-convex (or strongly midconvex) with modulus \(C(\cdot )\). If the above inequality holds for a fixed number \(\lambda \in (0,1)\), then we say that the process \(X\) is strongly \(\lambda\)-convex with modulus \(C(\cdot)\). The main subject of this paper is to extend some well-known results (such as the Jensen and Hermite-Hadamard inequalites) concerning convex functions to strongly convex stochastic processes. The Jensen-type theorem and Hermite-Hadamard-type theorem for strongly convex stochastic processes are given below. Theorem 1. Let \(X: I\times \Omega \rightarrow \mathbb{R}\) be a strongly convex stochastic process with modulus \(C(\cdot )\). Then \[ X\left( \sum_{i=1}^n \lambda_i t_i, \cdot \right) \leq \sum_{i=1}^n \lambda_i X(t_i, \cdot) -C(\cdot) \sum_{i=1}^n \lambda_i (t_i-\overline{t})^2 \qquad \text{(a.e.)} \] for all \(t_1, \dots , t_n \in I\), \(\lambda_1, \dots , \lambda_n >0\), such that \(\lambda_1 + \cdots +\lambda_n=1\) and \(\overline{t}=\lambda_1 t_1 + \cdots +\lambda_n t_n\). Theorem 2. Let \(X: I\times \Omega \rightarrow \mathbb{R}\) be a stochastic process which is strongly Jensen-convex with modulus \(C(\cdot )\) and mean-square continuous in the interval \(I\). Then, for any \(u,v \in I\), we have \[ \begin{multlined} X\left( \frac{u+v}{2}, \cdot \right) +C(\cdot) \frac{(v-u)^2}{12} \\ \leq \frac{1}{v-u} \int_u^v X(t, \cdot ) dt \leq \frac{X(u, \cdot )+X(v, \cdot)}{2} -C(\cdot) \frac{(u-v)^2}{6} \qquad \text{(a.e.)}. \end{multlined} \] Furthermore, the following properties are also proved: (i) If \(X\) is a strongly \(\lambda\)-convex stochastic process with modulus \(C(\cdot )\), then \(X\) is Jensen-convex with modulus \(C(\cdot )\). (ii) If \(X\) is a strongly Jensen-convex stochastic process with modulus \(C(\cdot )\) and \(\operatorname P\)-upper bounded on the interval \((a,b) \subset I\), then it is continuous in the interval \(I\). (iii) Let \(I\) be an open interval. A strongly Jensen-convex stochastic process \(X\) with modulus \(C(\cdot )\) is continuous if and only if it is strongly convex with modulus \(C(\cdot)\).