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The numerical analysis of stochastic time fractional evolution equationspresents considerable challenges due to the limited regularity of the modelcaused by the nonlocal operator and the presence of noise.  The existing time-stepping methods exhibit a significantly low orderconvergence rate. In this work, we introduce a smoothing technique and developthe novel high-order schemes for solving the linear stochastic fractionalevolution equations driven by integrated additive noise. Our approach involvesregularizing the additive noise through an $m$-fold integral-differentialcalculus, and discretizing the equation using the $k$-step BDF convolutionquadrature. This novel method, which we refer to as the ID$m$-BDF$k$ method, isable to achieve higher-order convergence in solving the stochastic models. Ourtheoretical analysis reveals that the convergence rate of the ID$2$-BDF2 methodis $O(\tau^{\alpha + \gamma -1/2})$ for $1< \alpha + \gamma \leq 5/2$, and$O(\tau^{2})$ for $5/2< \alpha + \gamma <3$, where $\alpha \in (1, 2)$ and$\gamma \in (0, 1)$ denote the time fractional order and the order of theintegrated noise, respectively. Furthermore, this convergence rate could beimproved to $O(\tau^{\alpha + \gamma -1/2})$ for any $\alpha \in (1, 2)$ and$\gamma \in (0, 1)$, if we employ the ID$3$-BDF3 method. The argument could beeasily extended to the subdiffusion model with $\alpha \in (0, 1)$. Numericalexamples are provided to support and complement the theoretical findings.

High-order BDF convolution quadrature for stochastic fractional evolution equations driven by integrated additive noise