  The combination of the It\^o formula and the Bismut-Elworthy-Li formula
implies that suitable smooth solutions of semilinear Kolmogorov partial
differential equations (PDEs) are also solutions to certain stochastic fixed
point equations (SFPEs). In this paper we generalize known results on existence
and uniqueness of solutions of SFPEs associated with PDEs with Lipschitz
continuous, gradient-independent nonlinearities to the case of
gradient-dependent nonlinearities. The main challenge arises from the fact that
in the case of a non-differentiable terminal condition and a gradient-dependent
nonlinearity the Bismut-Elworthy-Li formula leads to a singularity of the
solution of the SFPE in the last time point.
