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Abstract This paper deals with the existence of solution for an impulsive Riemann–Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1<β<2 $1<\beta<2$ and its Hyers–Ulam stability. We prove the mild solutions for the equation using basic theorems of fractional differential equation. The existence result of the equation is obtained by Mönch’s fixed point theorem. Finally, we prove the Hyers–Ulam stability of the solution.