  Upon the recent development of the quasi-reversibility method for terminal
value parabolic problems in \cite{Nguyen2019}, it is imperative to investigate
the convergence analysis of this regularization method in the stochastic
setting. In this paper, we positively unravel this open question by focusing on
a coupled system of Dirichlet reaction-diffusion equations with additive white
Gaussian noise on the terminal data. In this regard, the approximate problem is
designed by adding the so-called perturbing operator to the original problem
and by exploiting the Fourier reconstructed terminal data. By this way,
Gevrey-type source conditions are included, while we successfully maintain the
logarithmic stability estimate of the corresponding stabilized operator, which
is necessary for the error analysis. As the main theme of this work, we prove
the error bounds for the concentrations and for the concentration gradients,
driven by a large amount of weighted energy-like controls involving the
expectation operator. Compared to the classical error bounds in $L^2$ and $H^1$
that we obtained in the previous studies, our analysis here needs a higher
smoothness of the true terminal data to ensure their reconstructions from the
stochastic fashion. Two numerical examples are provided to corroborate the
theoretical results.
