  We obtain upper and lower Gaussian-type bounds on the density of each
component $Y^i_t$ of the solution $Y_t$ to a multidimensional non-Markovian
backward SDE. Our approach is based on the Nourdin-Viens formula and a
stochastic version of Wazewski's theorem on the positivity of the components of
a solution to an ODE. Furthermore, we apply our results to stochastic gene
expression; namely, we estimate the density of the law of the amount of protein
generated by a gene in a gene regulatory network.
