Mean-square Riemann-Stieltjes integrals of fuzzy stochastic processes and their applications (Q1971873)

The author studies a stochastic process \(X(t)\) with values in the space of upper semicontinuous \([0,1]\)-valued functions \(u\) on the Euclidean space that have maximum at \(1\), satisfy \(u(tx+ (1- t)y)\geq \min(u(x), u(y))\) for all \(x\), \(y\) from the domain of definition and \(t\in[0, 1]\), and such that \(\{x: u(x)> 0\}\) has a compact closure. Such a stochastic process is called a fuzzy stochastic process. The author modifies the standard definition of a Riemann-Stieltjes integral to integrate \(X(t)\) with respect to a numerical function \(g\) and to integrate \(g\) with respect to \(X(t)\) (under further assumptions on \(X(t)\)). Integrability conditions and a number of properties of such integral are established, most of them resembling the standard formulations and having straightforward proofs.