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Upon the recent development of the quasi-reversibility method for terminalvalue parabolic problems in \cite{Nguyen2019}, it is imperative to investigatethe convergence analysis of this regularization method in the stochasticsetting. In this paper, we positively unravel this open question by focusing ona coupled system of Dirichlet reaction-diffusion equations with additive whiteGaussian noise on the terminal data. In this regard, the approximate problem isdesigned by adding the so-called perturbing operator to the original problemand by exploiting the Fourier reconstructed terminal data. By this way,Gevrey-type source conditions are included, while we successfully maintain thelogarithmic stability estimate of the corresponding stabilized operator, whichis necessary for the error analysis. As the main theme of this work, we provethe error bounds for the concentrations and for the concentration gradients,driven by a large amount of weighted energy-like controls involving theexpectation 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 highersmoothness of the true terminal data to ensure their reconstructions from thestochastic fashion. Two numerical examples are provided to corroborate thetheoretical results.

An improved quasi-reversibility method for a terminal-boundary value multi-species model with white Gaussian noise