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We consider equidistant approximations of stochastic integrals driven byH\"older continuous Gaussian processes of order $H>\frac12$ with discontinuousintegrands involving bounded variation functions. We give exact rate ofconvergence in the $L^1$-distance and provide examples with different drivers.It turns out that the exact rate of convergence is proportional to $n^{1-2H}$that is twice better compared to the best known results in the case ofdiscontinuous integrands, and corresponds to the known rate in the case ofsmooth integrands. The novelty of our approach is that, instead of usingmultiplicative estimates for the integrals involved, we apply change ofvariables formula together with some facts on convex functions allowing us tocompute expectations explicitly.

On sharp rate of convergence for discretisation of integrals driven by  fractional Brownian motions and related processes with discontinuous  integrands