New generalized mean square stochastic fractional operators with applications

This paper aims to develop novel generalized mean square fractional integral and derivative operators (for second-order stochastic processes) and presents an application of these operators. To conduct our study, first, the Riemann-Liouville’s approach is applied to iterate and fractionalize two specially constructed mean square Riemann integral operators. This procedure leads us to define new generalized left-sided and right-sided mean square fractional integral operators based on which the associated generalized mean square fractional derivative operators (left- and right-sided) are also defined. Some properties such as linearity, semigroup property, boundedness, continuity, and inverse property are investigated for these operators. It is proved that new classes of these left- and right-sided integral and derivative operators constitute Banach spaces and they also exist in the sense of probability. These operators generalize various mean square operators of the types Katugampola, Hadamard, Riemann-Liouville, Riemann, in which the operators of Riemann-Liouville and that of the Riemann have already been previously defined and the rest are new in the sense of mean square stochastic calculus. As an application, the newly established results are applied to prove a generalized fractional stochastic version of the well-known Hermite-Hadamard inequality for convex stochastic processes, where a bound for the absolute of the difference between the two rightmost terms is also obtained. Some relations of our results with the already existing results are also discussed.