On a discrete fractional stochastic Gronwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale

The aim of this paper is to derive a novel discrete form of stochastic fractional Gronwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Gronwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < alpha < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.